Abstract

We propose two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and an asymptotically κ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions.

1. Introduction

Let be a real Hilbert space with inner product and norm , let be a nonempty closed convex subset of , and let be the metric projection of onto . Let be a self-mapping on . is the set of fixed points of and is the set of all real numbers. We recall that a mapping is said to be -Lipschitz continuous, if there exists a constant such that If , then is called a nonexpansive mapping and if , then is called a contraction. We also let denote the identity operator on the Hilbert space .

Let be a real-valued function, let be a nonlinear mapping, and let be a bifunction. The generalized mixed equilibrium problem (GMEP) introduced in [1] is to find such that We denote the set of solutions of GMEP (2) by . The GMEP covers many problems [2–6] as special cases and has been extensively studied recently.

Throughout this paper, we assume as in [1] that is a bifunction satisfying conditions (A1)–(A4) and is a lower semicontinuous and convex function with restriction (B1) or (B2), where(A1), for all ;(A2) is monotone; that is, , for any ;(A3) is upper-hemicontinuous; that is, for each , (A4) is convex and lower semicontinuous, for each ;(B1)for each and , there exists a bounded subset   and such that, for any , (B2) is a bounded set.

Next we list some elementary consequences for the mixed equilibrium problem studied in [2] where is the solution set.

Proposition 1 (see [2]). Assume that satisfies (A1)–(A4) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows: for all . Then the following hold:(i)for each ;(ii) is single-valued;(iii) is firmly nonexpansive; that is, for any , (iv);(v) is closed and convex.

Next, recall some concepts.

Definition 2. Let be a nonempty subset of a normed space and let be a self-mapping on .(i) is asymptotically nonexpansive (see [7]), if there exists a sequence of positive numbers satisfying the property and (ii) is asymptotically nonexpansive in the intermediate sense (see [8]) provided that is uniformly continuous and (iii) is uniformly Lipschitzian, if there exists a constant such that

It is clear that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is uniformly Lipschitzian. Recently, Kim and Xu [9] introduced the concept of asymptotically -strict pseudocontractive mappings in a Hilbert space as follows.

Definition 3. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping with sequence , if there exist a constant and a sequence in with such that
They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically -strict pseudocontractive mapping with sequence is a uniformly -Lipschitzian mapping with .

Recently, Sahu et al. [10] considered the concept of asymptotically -strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 4. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence , if there exist a constant and a sequence in with such that
Put . Then, , , and (11) reduces to the relation
Whenever , for all in (12), then is an asymptotically -strict pseudocontractive mapping with sequence .

Let be a single-valued mapping of into and be a multivalued mapping with domain . Consider the following variational inclusion: find a point such that We denote by the solution set of the variational inclusion (13). In 1998, Huang [11] studied problem (13) in the case where is maximal monotone and is strongly monotone and Lipschitz continuous with . Subsequently, Zeng et al. [12] further studied problem (13) in the case which is more general than that in [11]. Moreover, the authors [12] obtained the same strong convergence conclusion as in [11]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions.

In this paper, inspired by the research work mentioned above, we introduce two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions.

2. Preliminaries

Let be a real Hilbert space and be a nonempty closed convex subset of . We use the notation to indicate that the sequence converges weakly to and to indicate that the sequence converges strongly to . Moreover, we use to denote the weak -limit set of the sequence ; that is,

Recall that a mapping is called(i)Monotone, if (ii)-strongly monotone, if there exists a constant such that (iii)-inverse strongly monotone, if there exists a constant such that

When is an -inverse strongly monotone mapping of into , it is easy to see that is -Lipschitz continuous. We also have that, for all and , So, if , then is a nonexpansive mapping from to .

The metric projection from onto is the mapping which assigns to each point the unique point satisfying the property

Some important properties of projections are listed in the following proposition.

Proposition 5. For given and ,(i), ;(ii), ;(iii), .
Consequently, is nonexpansive and monotone. It is easy to see that the projection is -ism.

We need some facts and tools in a real Hilbert space which are listed as lemmas below.

Lemma 6. Let be a real inner product space. Then there holds the following inequality:

Lemma 7. Let be a real Hilbert space. Then the following hold:(a), for all ;(b), for all and with ;(c)if is a sequence in such that , it follows that

Lemma 8 (see [10, Lemma 2.5]). Let be a real Hilbert space. Given a nonempty closed convex subset of and points and given also a real number , the set is convex and closed.

Lemma 9 (see [10, Lemma 2.6]). Let be a nonempty subset of a Hilbert space and let be an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then for all and .

Lemma 10 (see [10, Lemma 2.7]). Let be a nonempty subset of a Hilbert space and let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Let be a sequence in such that and as . Then as .

Lemma 11 (demiclosedness principle [10, Proposition 3.1]). Let be a nonempty closed convex subset of a Hilbert space and let be a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Then is demiclosed at zero in the sense that if is a sequence in such that and , then .

Lemma 12 (see [10, Proposition 3.2]). Let be a nonempty closed convex subset of a Hilbert space and let be a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that . Then, is closed and convex.

Lemma 13 (see [13, page 80]). Let , and be sequences of nonnegative real numbers satisfying the inequality If and , then exists. If, in addition, has a subsequence which converges to zero, then .

Corollary 14 (see [14, page 303]). Let and be two sequences of nonnegative real numbers satisfying the inequality If converges, then exists.

Recall that a Banach space is said to satisfy the Opial condition [15], if for any given sequence which converges weakly to an element , there holds the inequality It is well known in [15] that every Hilbert space satisfies the Opial condition.

Lemma 15 (see [16, Proposition 3.1]). Let be a nonempty closed convex subset of a real Hilbert space and let be a sequence in . Suppose that where and are sequences of nonnegative real numbers such that and . Then converges strongly in .

Lemma 16 (see [17]). Let be a closed convex subset of a real Hilbert space . Let be a sequence in and . Let . If is such that and satisfies the condition then as .
Let be an infinite family of nonexpansive self-mappings on and let be a sequence of nonnegative numbers in . For any , define a self-mapping on as follows: Such a mapping is called the -mapping generated by and .

Lemma 17 (see [18, Lemma 3.2]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then, for every and , the limit exists.

Utilizing Lemma 17, we can define a mapping as follows: Such a is called the -mapping generated by and . We remark that since is nonexpansive, is also nonexpansive.

Lemma 18 (see [18, Lemma 3.3]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that , and let be a sequence in , for some . Then, .

Lemma 19 (see [19, demiclosedness principle]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive self-mapping on . Then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that .

Recall that a set-valued mapping is called monotone, if, for all , and , we have A set-valued mapping is called maximal monotone, if is monotone and , for each . We denote by the graph of . It is known that a monotone mapping is maximal if and only if, for , for every , we have .

Assume that is a maximal monotone mapping. Then, for , associated with , the resolvent operator can be defined as We have the following property for the resolvent operator .

Lemma 20 (see [11]). is single-valued and firmly nonexpansive; that is,

Consequently, is nonexpansive and monotone.

Lemma 21 (see [20]). Let be a maximal monotone mapping with . Then, for any given , is a solution of the variational inclusion: find a point such that
where is a single-valued mapping of into if and only if satisfies

Lemma 22 (see [12]). Let be a maximal monotone mapping with and let be a strongly monotone, continuous, and single-valued mapping. Then, for each , the equation has a unique solution  , for .

Lemma 23 (see [20]). Let be a maximal monotone mapping with and let be a monotone, continuous, and single-valued mapping. Then , for each . In this case, is maximal monotone.

3. Strong Convergence Theorem

In this section, we prove a strong convergence theorem for an iterative algorithm for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. This iterative algorithm is based on the extragradient method [21], Mann-type iterative method, and shrinking projection method. For more recent related results, see [22] and the references therein.

Theorem 24. Let be a nonempty closed convex subset of a real Hilbert space . Let be two integers. Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function, where . Let be a maximal monotone mapping and let and be -inverse strongly monotone and -inverse strongly monotone, respectively, where and . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense, for some , with sequence such that and such that . Let be a sequence of nonexpansive self-mappings on and let be a sequence in for some . Assume that is nonempty and bounded and that either (B1) or (B2) holds. Let , and be sequences in such that , , and . Pick any and set . Let be a sequence generated by the following algorithm: where is the -mapping generated by (2.2), , and . Assume that the following conditions hold:(i);(ii);(iii).Then converges strongly to .

Proof. We divide the proof into several steps.
Step  1. First note that the defining inequality in is equivalent to the inequality So, by Lemma 8, is closed and convex, for every .
We next show that , for all . Put for all and ; for all and , and , where is the identity mapping on . Then we have that and . Suppose that for some . Take arbitrarily. Then, from (18) and Proposition 1(iii), we have Similarly, we have Combining (40) and (41), we have By Lemma 7(b), we deduce from (36) and (42) that It follows from (42), (43), and the convexity of that Hence, . This implies that , for all .
Step  2. We prove that as .
Indeed, let . From and , we obtain This implies that is bounded and, hence, , , , and are also bounded. Since and , we have Therefore, exists. From , by Proposition 5(ii) we obtain which implies It follows from that and hence From (48) and , we have Also, utilizing Lemma 7(b), we obtain from (36), (42), and (43) that which, hence, yields Since and are bounded, it follows from (50) and condition (i) that Note that which leads to So, from (50), (53), and , we get
Step  3. We prove that , , , and as .
Indeed, from (42) and (43), it follows that Next, we prove that For , it follows from (18) that By (40), (41), (57), and (59), we obtain which implies that Since , , , and (56), we have By Proposition 1(iii) and Lemma 7(a), we have which implies that Combining (57) and (64), we have which implies From , (56), and (62), we know that (58) holds. Hence, we obtain
Next, we show that , . It follows from Lemma 20 and (18) that Combining (57) and (68), we have which implies From , , , and (56), we obtain By Lemmas 20 and 7(a), we obtain which implies Combining (57) and (73), we get which implies From (56), (71), and , we have From (76), we get By (67) and (77), we have From (48) and (78), we have By (56), (67), and (77), we get We observe that From and (80), we have We note that From (79), (82), and Lemma 9, we obtain On the other hand, we note that From (82), (84), and the uniform continuity of , we have In addition, note that So, from (53), (78), and [4, Remark 3.2], it follows that
Step  4. Finally we prove that as .
Indeed, since is bounded, there exists a subsequence which converges weakly to some . From (58) and (76)–(78), we have that and , where and . Since is uniformly continuous, by (86), we get , for any . Hence, from Lemma 11, we obtain . In the meantime, utilizing Lemma 19, we deduce from (88) and that . Next, we prove that . As a matter of fact, since is -inverse strongly monotone, is a monotone and Lipschitz continuous mapping. It follows from Lemma 23 that is maximal monotone. Let ; that is, . Again, since , , we have that is, In terms of the monotonicity of , we get and, hence, In particular, Since (due to (76)) and (due to the Lipschitz continuity of ), we conclude from and condition (ii) that It follows from the maximal monotonicity of that ; that is, . Therefore, . Next, we prove that . Since , , , we have By (A2), we have Let , for all and . This implies that . Then, we have By (58), we have as . Furthermore, by the monotonicity of , we obtain . Then, by (A4), we obtain Utilizing (A1), (A4), and (98), we obtain and, hence, Letting , we have, for each , This implies that and, hence, . Consequently, . This shows that . From (45) and Lemma 16, we infer that as . This completes the proof.

4. Weak Convergence Theorem

In this section, we prove a weak convergence theorem for an iterative algorithm for finding a common element of the set of solutions of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. This iterative algorithm is based on the extragradient method and Mann-type iterative method.

Theorem 25. Let be a nonempty closed convex subset of a real Hilbert space . Let be two integers. Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function, where . Let be a maximal monotone mapping and let and be -inverse strongly monotone and -inverse strongly monotone, respectively, where , . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense, for some , with sequences and . Let be a sequence of nonexpansive self-mappings on and be a sequence in , for some . Assume that is nonempty and that either (B1) or (B2) holds. Let , and be sequences in such that and . Pick any and let be a sequence generated by the following algorithm: where is the -mapping generated by (2.2). Assume that the following conditions hold:(i) and ;(ii);(iii), ;(iv), .Then, converges weakly to .

Proof. First, let us show that exists, for any . Put for all , ; for all , , , where is the identity mapping on . Then, we have that and . Take arbitrarily. Similar to the proof of Theorem 24, we obtain that We observe that It follows from (105), (106), and (111) that By Lemma 13 and condition (i), we deduce that exists. Hence, is bounded and so are , , and .
In addition, by Lemma 7(b), we obtain from (105), (106), and (111) that which immediately yields From , , , and the existence of , it follows that Again, utilizing Lemma 7(b), we obtain from (105), (106), and (111) that which leads to From , , condition (ii), and the existence of , it follows that Note that Hence, it is easy to see from that From (115) and (118), it follows that Combining (107) and (111), we have where , which implies From , , , , and (121), we have Combining (108) and (111), we have which implies From , (121), and (124), we get From (127), we have Combining (109) and (111), we obtain where , which implies From , , , , and (121), we get Combining (110) and (111), we have which implies From , , (121), and (131), we obtain By (134), we have From (128) and (135), we have By (121) and (136), we obtain We note that From and (137), we have On the other hand, we observe that By (115) and (136), we have We note that From (139), (141), Lemma 9, and the uniform continuity of , we obtain In addition, note that So, from (118), (136), and [4, Remark 3.2], it follows that
Since is bounded, there exists a subsequence of which converges weakly to . From (136), we have that . From (143) and the uniform continuity of , we have , for any . So, from Lemma 11, we have . Utilizing the similar arguments to those in the proof of Theorem 24, we can derive . Consequently, . This shows that .
Next, let us show that is a single-point set. As a matter of fact, let be another subsequence of such that . Then, we get . If , from the Opial condition, we have This attains a contradiction. So we have . Put . Since , we have . By Lemma 15, we have that converges strongly to some . Since converges weakly to , we have Therefore, we obtain . This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant no. 30-130-35-HiCi. The authors, therefore, acknowledge the technical and financial support of KAU. The authors thank the referees for their valuable comments and appreciation.