A hybrid projection method for solving a common solution of a system of equilibrium problems and fixed point problems for asymptotically strict pseudocontractions in the intermediate sense in Hilbert spaces

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.

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 α-inverse-strongly monotone[1,2], if there exists a positive real number α such that

Clearly, if A is α-inverse-strongly monotone, then A is monotone.

A mapping A is called β-strongly monotone if there exists a positive real number β such that

A mapping A is called L-Lipschitz continuous if there exists a positive real number L such that

It is easy to see that if A is an α-inverse-strongly 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 asymptoticallyκ-strictly 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 CQ-algorithm 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 anα-inverse-strongly monotone mapping. Let for each, be a uniformly continuous-strictly 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 uniformlyL-Lipschitz 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.

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 w-limit set of .

(3) A nonlinear mapping S : is a self-mapping 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 self-mapping 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 self-mapping 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 λ-strict 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 asymptoticallyλ-strict 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 asymptoticallyλ-strict 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:

(A1) for all ;

(A2) F is monotone, i.e., , for all ;

(A3) for each , ;

(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:

(1) is single-valued;

(2) is firmly nonexpansive, i.e., for any,

(3) ; and

(4) is closed and convex.

Lemma 2.5 ([7,19])

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, andbe a uniformlyL-Lipschitz continuous and asymptoticallyλ-strict 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 uniformlyL-Lipschitz continuous and asymptoticallyλ-strict 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

Then.

Lemma 2.9 ([4])

LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe an asymptoticallyλ-strict pseudocontractive mapping in the intermediate sense with the sequence. Then

for alland.

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 asymptotically-strict 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 , .

Since and , we have

Therefore, , and so

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

Since , we have

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

This implies that for each ,

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

for any , which gives that

Moreover, for each , we obtain

This implies that

Step 6. We will show that .

(6.1) We will show 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

(6.2) We will show that .

By Lemma 2.3, for each , we have

From (A2), we get

Taking , 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

Letting , from (A3), we get

Therefore, , and so .

Step 7. We will show that converges strongly to .

Set , then

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. □

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 asymptotically-strict 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 asymptotically-strict 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].

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

where .

It is easy to see that is discontinuous at and S is not Lipschitz continuous.

Set and .

For each , we have

For each , we have

For each and , we have

It follows that

for all and and for some .

Therefore, S is an asymptotically k-strict 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 one-dimensional 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.

The authors declare that they have no competing interests.

All authors contributed equally and significantly in this research. All authors read and approved the final manuscript.

This research was supported by the Faculty of Science, KMUTT Research Fund 2553-2554.

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