Abstract

We introduce two iterative algorithms by the hybrid extragradient method with regularization for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings and the set of fixed points of 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.

1. Introduction

Throughout this paper, we assume that is 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 . We denote by the set of fixed points of and by the set of all real numbers.

Let be a real-valued function, let be a nonlinear mapping, and let be a bifunction. In 2008, Peng and Yao [1] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that We denote the set of solutions of GMEP (1) by . The GMEP (1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems in noncooperative games. The GMEP is further considered and studied in [25].

Let be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing over the constraint set (assuming the existence of minimizers). We denote by the set of minimizers of CMP (2). The gradient-projection algorithm (GPA) generates a sequence determined by the gradient and the metric projection as follows: or more generally, where, in both (3) and , the initial guess is taken from arbitrarily and the parameters or are positive real numbers. The convergence of algorithms (3) and depends on the behavior of the gradient .

Since the Lipschitz continuity of the gradient implies that it is actually -inverse strongly monotone (ism) [6], its complement can be an averaged mapping (i.e., it can be expressed as a proper convex combination of the identity mapping and a nonexpansive mapping). Consequently, the GPA can be rewritten as the composite of a projection and an averaged mapping, which is again an averaged mapping. This shows that averaged mappings play an important role in the GPA. Recently, Xu [7] used averaged mappings to study the convergence analysis of the GPA, which is hence an operator-oriented approach.

Assume that the CMP (2) is consistent and the gradient is -Lipschitz continuous with . Let be a -contraction with . Xu [7] introduced the following hybrid GPA: where and . It was proven that under appropriate conditions the sequence converges in norm to a minimizer of CMP (2); see [7, Theorem 5.2].

It is worth emphasizing that the regularization, in particular the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. Consider the regularized minimization problem where is the regularization parameter and again is convex with -Lipschitz continuous gradient . In [7], Xu introduced another hybrid GPA with regularization where (i) for all ; (ii) (and ) as ; (iii) ; and (iv) as . It was proven that converges strongly to the minimum-norm solution of CMP (2); see [7, Theorem 6.1]. Very recently, the hybrid GPA with regularization is extended to develop new extragradient methods with regularization in Ceng et al. [8, 9] for finding a common solution of the split feasibility problem (SFP) and the fixed point problem of a nonexpansive mapping in a real Hilbert space.

On the other hand, consider the following variational inequality problem (VIP): find a such that The solution set of VIP (7) is denoted by .

The VIP (7) was first discussed by Lions [10] and now is well known; there are a lot of different approaches towards solving VIP (7) in finite-dimensional and infinite-dimensional spaces, and the research is intensively continued. The VIP (7) has many applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and other fields; see, for example, [1114]. It is well known that if is a strongly monotone and Lipschitz-continuous mapping on , then VIP (7) has a unique solution. Not only the existence and uniqueness of solutions are important topics in the study of VIP (7), but also how to actually find a solution of VIP (7) is important.

Motivated by the idea of Korpelevič's extragradient method [15], Nadezhkina and Takahashi [16] introduced an extragradient iterative scheme: where is a monotone, -Lipschitz continuous mapping, is a nonexpansive mapping, and for some and for some . They proved the weak convergence of to an element of . Recently, inspired by Nadezhkina and Takahashi's iterative scheme [16], Zeng and Yao [17] introduced another iterative scheme for finding an element of and derived the weak convergence result. Furthermore, by combining the CQ method and extragradient method, Nadezhkina and Takahashi [18] introduced an iterative process: They proved the strong convergence of to an element of under appropriate conditions. Later on, Ceng and Yao [19] introduced an extragradient-like approximation method which is based on the above extragradient method and viscosity approximation method and derived a strong convergence result as well. Next, recall some concepts. A mapping is called -Lipschitz continuous if there exists a constant such that In particular, if , then is called a nonexpansive mapping; if , then is called a contraction.

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

It is obvious that if is -inverse-strongly monotone, then is monotone and -Lipschitz continuous.

Definition 1. Let be a nonempty subset of a normed space and let be a self-mapping on .(i) is asymptotically nonexpansive (cf. [20]) if there exists a sequence of positive numbers satisfying the property and (ii) is asymptotically nonexpansive in the intermediate sense [21] provided 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.

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [20] as an important generalization of the class of nonexpansive mappings. The existence of fixed points of asymptotically nonexpansive mappings was proved by Goebel and Kirk [20] as follows.

Theorem GK (see [20, Theorem ]). If is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive mapping has a fixed point in .

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [21]. Recently, Kim and Xu [22] introduced the concept of asymptotically -strict pseudocontractive mappings in a Hilbert space as follows.

Definition 2. Let be a nonempty subset of a Hilbert space . A mapping is said to be an asymptotically -strict pseudocontractive mapping with sequence if there exists 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. [23] considered the concept of asymptotically -strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 3. 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 (17) reduces to the relation

Whenever for all in (18), then is an asymptotically -strict pseudocontractive mapping with sequence . In 2009, Sahu et al. [23] derived the weak and strong convergence of the modified Mann iteration process for the class of asymptotically -strictly pseudocontractive mappings in the intermediate sense with sequence . More precisely, they established the following theorems.

Theorem SXY1. Let be a nonempty closed convex subset of a real Hilbert space and let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that and . Assume that is a sequence in such that and . Let be a sequence in generated by the modified Mann iteration process: Then converges weakly to an element of .

Theorem SXY2. Let be a nonempty closed convex subset of a real Hilbert space and let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence such that is nonempty and bounded. Let be a sequence in such that for all . Let be the sequence in generated by the following (CQ) algorithm: where and . Then converges strongly to .

Subsequently, the iterative algorithms in Theorems SXY1 and SXY2 are extended to develop new iterative algorithms for finding a common solution of the VIP and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space; see, for example, [24, 25].

On the other hand, Yao et al. [26] introduced two iterative algorithms for finding a common element of the set of fixed points of an asymptotically -strict pseudocontraction and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then they obtained some weak and strong convergence theorems for the proposed iterative algorithms. Very recently, motivated by Yao et al. [26], Cai and Bu [3] introduced 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 inequalities for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Then they proved some strong and weak convergence theorems for the proposed iterative algorithms under appropriate conditions.

In this paper, inspired by the above facts, we introduce two iterative algorithms by hybrid extragradient method with regularization for finding a common element of the set of solutions of the CMP (2) for a convex functional with -Lipschitz continuous gradient , the set of solutions of finite GMEPs, the set of solutions of finite VIPs for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Then we prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions. For recent related results, see, for example, [7, 24, 2731] and ther references therein.

2. Preliminaries

Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of . We write 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,

The metric (or nearest point) projection from onto is the mapping which assigns to each point the unique point satisfying the property Some important properties of projections are gathered in the following proposition.

Proposition 4. For given and :(i),   for all  ;(ii),  for all  ;(iii),  for all  .

Consequently, is nonexpansive and monotone.

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

Definition 5. A mapping is said to be(a)nonexpansive if (b)firmly nonexpansive if is nonexpansive, or equivalently, if is -inverse strongly monotone (-ism), alternatively, is firmly nonexpansive if and only if can be expressed as where is nonexpansive; projections are firmly nonexpansive.

It can be easily seen that if is nonexpansive, then is monotone. It is also easy to see that a projection is -ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.

Definition 6. A mapping is said to be an averaged mapping if it can be written as the average of the identity and a nonexpansive mapping; that is, where and is nonexpansive. More precisely, when the last equality holds, we say that is -averaged. Thus, firmly nonexpansive mappings (in particular, projections) are -averaged maps.

Proposition 7 (see [32]). Let   be a given mapping.(i) is nonexpansive if and only if the complement is -ism.(ii)If is -ism, then for is -ism.(iii) is averaged if and only if the complement is -ism for some . Indeed, for is -averaged if and only if is -ism.

Proposition 8 (see [32, 33]). Let be given operators.(i)If for some and if is averaged and is nonexpansive, then is averaged.(ii) is firmly nonexpansive if and only if the complement is firmly nonexpansive.(iii)If for some and if is firmly nonexpansive and is nonexpansive, then is averaged.(iv)The composite of finitely many averaged mappings is averaged. That is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then the composite is -averaged, where .(v)If the mappings are averaged and have a common fixed point, thenThe notation denotes the set of all fixed points of the mapping ; that is, = .

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

Lemma 9. Let be a real inner product space. Then there holds the following inequality:

Lemma 10. Let be a bounded sequence on a reflexive Banach space . If , then .

Lemma 11. Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 4 (i)) implies

Lemma 12. Let be a real Hilbert space. Then the following hold:(i) for all ;(ii) for all and for all ;(iii)If is a sequence in such that , it follows that

Lemma 13 ([23, 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 14 ([23, 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 15 ([23, 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 16 (demiclosedness principle [23, 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 17 ([23, 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.

Remark 18. Lemmas 16 and 17 give some basic properties of an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Moreover, Lemma 16 extends the demiclosedness principles studied for certain classes of nonlinear mappings in Kim and Xu [22], Górnicki [34], Marino and Xu [35], and Xu [36].

To prove a weak convergence theorem by a modified extragradient method with regularization for the CMP (2) and the fixed point problem of an asymptotically -strict pseudocontractive mapping in the intermediate sense, we need the following lemma due to Osilike et al. [37].

Lemma 19 (see [37, 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 20 (see [38, 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 [39] if for any given sequence which converges weakly to an element , there holds the inequality It is well known in [39] that every Hilbert space satisfies the Opial condition.

Lemma 21 (see [24, 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 .

A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if its graph is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for for all implies . Let be a monotone, -Lipschitz continuous mapping, and let be the normal cone to at ; that is, . Define It is known that in this case is maximal monotone, and if and only if ; see [40].

For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions:(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.

Lemma 22 (see [41]). 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:(1)for each ;(2) is single-valued;(3) is firmly nonexpansive; that is, for any , (4);(5) is closed and convex.

Lemma 23 (see [42]). 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 .

3. Strong Convergence Theorem

In this section, we prove a strong convergence theorem for a hybrid extragradient iterative algorithm with regularization for finding a common element of the set of solutions of the CMP (2) for a convex functional with -Lipschitz continuous gradient , the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. This iterative algorithm with regularization is based on the extragradient method, shrinking projection method, Mann-type iterative method, and hybrid gradient projection algorithm (GPA) with regularization.

Theorem 24. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . 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 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 sequence such that and such that . Assume that is nonempty and bounded. Let be a sequence in and let be sequences in such that and . Pick any and set , . Let be a sequence generated by the following algorithm: where , and . Assume that the following conditions hold:(i);(ii) for some ;(iii),for all;(iv),for all.Then converge strongly to provided either (B1) or (B2) holds.

Proof. First of all, one can show that is -averaged for each , where which shows that is nonexpansive. Furthermore, for with , we have Without loss of generality, we may assume that Consequently, it follows that for each integer , is -averaged with This immediately implies that is nonexpansive for all .
We divide the proof into several steps.
Step 1. We show that is well defined. It is obvious that is closed and convex. As the defining inequality in is equivalent to the inequality by Lemma 13 we know that is convex for every .
Next we 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 (23) and Lemma 22 we have Similarly, we have Combining (52) and (53), we have Also, it follows from (44) that Note that for every . Then, by Proposition 4(ii), we have Further, by Proposition 4(i), we have So from (54) and (55), we obtain By Lemma 12 and (58), we have It follows from (59) and that Hence . This implies that for all . Therefore, is well defined.
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 , we obtain which implies It follows from that and hence From (64) and , we have Note that Since and (66), we obtain
Step 3. We prove that , and as .
Indeed, from (58) and (59) it follows that Next we prove that For , it follows from (23) that By (52), (53), (58), (69), and (71), we obtain which implies that Since , , , , and , we conclude from (68) and the boundedness of that By Lemmas 12 and 22, we have which implies that Utilizing (72) and (76), we have which implies Since , , , , , and , we conclude from (68) and (74) and the boundedness of that (70) holds. Hence we obtain
Next we show that , . It follows from (23) that Utilizing (54), (72), and (80), we get which implies Since , , , , , and , we conclude from (68) and the boundedness of that By Proposition 4 and Lemma 12, we obtain which implies Utilizing (81) and (85), we get which implies Since , , , , , and , we conclude from (68) and (83) and the boundedness of , and that From (88) we get By (79) and (89), we have From (64) and (90), we have By (68), (79), and (89), we get On the other hand, utilizing (58) and (59) we have which yields Since , , and , we conclude from (92) and the boundedness of , and that Also, utilizing the similar arguments to those of (58), we obtain which together with (59) leads to So we have Since , , and , we conclude from (92) and the boundedness of , and that Utilizing (92)–(99), we get Since , we get which together with (100) implies In addition, observe that From (68), (79), and (100), it immediately follows that Moreover, note that From (91), (92), and (100), it immediately follows that Meantime, it is clear that From (102) and (106) and Lemma 14, we obtain Furthermore, we note that From (102) and (108) and the uniform continuity of , we have
Step 4. Finally we prove that as .
Indeed, since is bounded, there exists a subsequence which converges weakly to some . From (70), (88)–(90), and (104), we have that ,    ,    ,   ,  where . Since is uniformly continuous, by (110) we get for any . Hence from Lemma 16, we obtain . Next we prove that . Let where . Let . Since and , we have On the other hand, from and , we have and hence Therefore, we have From (88) and since is continuous, we obtain that . From , , for all and (88), we have Since is maximal monotone, we have and hence , , which implies . Next we prove that . Since , , , we have By (A2), we have Let for all and . This implies that . Then we have By (70), we have as . Furthermore, by the monotonicity of , we obtain . Then by (A4), we obtain Utilizing (A1), (A4), and (120), we obtain and hence Letting , we have, for each , This implies that and hence .
Further, let us show that . As a matter of fact, since and (due to (95) and (99)), we have  and . Let where is the normal cone to at . We have already mentioned that in this case the mapping is maximal monotone, and if and only if ; see [40] for more details. Let be the graph of and let . Then, we have and hence . So we have for all . On the other hand, from and , we have and hence Therefore, from for all and , we have Note that , and (due to the -Lipschitz continuity of ). Thus, we obtain as . Since is maximal monotone, we have and hence . Clearly, . Consequently, . This shows that . From (61) and Lemma 23 we infer that as . This completes the proof.

Corollary 25 (i.e., [3, Theorem 3.1]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . 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 and let 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 sequence such that and such that . Assume that is nonempty and bounded. Let and be sequences in such that and . Pick any and set , . Let be a sequence generated by the following algorithm: where , , , , , . Assume that either (B1) or (B2) holds. Then converge strongly to .

Proof. In Theorem 24, put and for all . Then and . In this case, we obtain from (44) that Thus, the iterative scheme (44) reduces to (128). Since for all and is bounded, we know that , and It is easy to see that all the conditions of Theorem 24 are satisfied. Therefore, in terms of Theorem 24, we derive the desired result.

Corollary 26. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. 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 sequence such that and such that . Assume that is nonempty and bounded. Let be a sequence in , and let be sequences in such that and . Pick any and set , . Let be a sequence generated by the following algorithm: where , and . Assume that the following conditions hold:(i);(ii) for some ;(iii), ;(iv).Then converge strongly to provided either (B1) or (B2) holds.

Corollary 27. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively. Let be a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequence such that and such that . Assume that is nonempty and bounded. Let be a sequence in , and let be sequences in such that and . Pick any and set . Let be a sequence generated by the following algorithm: where , , and . Assume that the following conditions hold:(i);(ii) for some ;(iii);(iv).Then converge strongly to provided either (B1) or (B2) holds.

Corollary 28. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively. Let be a uniformly continuous asymptotically -strict pseudocontractive mapping for some with sequence such that . Assume that is nonempty and bounded. Let be a sequence in , and let be sequences in such that and . Pick any and set . Let be a sequence generated by the following algorithm: where , , and . Assume that the following conditions hold:(i);(ii) for some ;(iii);(iv).Then converge strongly to provided either (B1) or (B2) holds.

Remark 29. Theorem 24 extends, improves, supplements, and develops Yao et al. [26, Theorem 3.1] in the following aspects.(i)Theorem 24 generalizes and extends [26, Theorem 3.1] from the asymptotically -strict pseudocontractive mapping to the asymptotically -strict pseudocontractive mapping in the intermediate sense and from one mixed equilibrium problem to finite GMEPs.(ii)We add finite VIPs and the CMP (2) in our algorithm such that it can be applied to find a common element of the set of solutions of finite GMEPs, the set of solutions of finite VIPs for inverse strongly monotone mappings, the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense, and the CMP (2) for a convex functional with -Lipschitz continuous gradient .(iii)Theorem 24 also removes the condition (ii) in [26, Theorem 3.1].

4. Weak Convergence Theorem

In this section, we prove a new weak convergence theorem by a modified extragradient method with regularization for finding a common element of the set of solutions of the CMP (2) for a convex functional with -Lipschitz continuous gradient , the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. This iterative algorithm with regularization is based on the extragradient method, Mann-type iterative method, and hybrid gradient projection algorithm (GPA) with regularization.

Theorem 30. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . 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 and be -inverse strongly monotone and -inverse-strongly monotone, respectively, where , . Let be the uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequences and . Assume that is nonempty. Let be a sequence in , and let be sequences in such that and . Pick any and let be a sequence generated by the following algorithm: where , , , and . Assume that either (B1) or (B2) holds and that the following conditions are satisfied:(i) and ;(ii) for some .Then(a) converges weakly to an element ;(b) converges weakly to provided is bounded.

Proof. First of all, again one can show that is -averaged for each , where This shows that is nonexpansive. Furthermore, for with , without loss of generality, we may assume that Consequently, it follows that for each integer , is -averaged with This immediately implies that is nonexpansive for all .
Next let us show that exists for any . Put for all , , and , where is the identity mapping on . Then we have that and . Take arbitrarily. Similarly to the proof of Theorem 24, we obtain that We observe that It follows from (142) and (147) and that From and condition (i) we have So, applying Lemma 19 to (148), we deduce that exists. This implies that is bounded and hence , and are also bounded. In addition, by Lemma 12 and (59) we obtain from that Thus, it is easy to see from that Since exists, , , and the sequence is bounded, we obtain Taking into consideration , we get , which together with (152) leads to Combining (140), (142), (143), and (147), we have which yields Since , , , , , and , we conclude from (152) and the boundedness of , and that Combining (140), (142), (144), and (147), we have which implies Since , , , , and , we conclude from (152) and (156) and the boundedness of , and that From (159), we have Combining (140), (142), (145), and (147), we have where , which implies Since , , , , , and , we conclude from (152) and the boundedness of , and that Combining (140), (142), (146), and (147), we have which implies Since , , , , , and , we conclude from (152) and (163) and the boundedness of , and that By (166), we have From (160) and (167), we have By (152) and (168), we obtain
Furthermore, combining (139), (140), (142), and (147), we have which yields Since , , , and , we obtain from (152) and the boundedness of , and that Also, combining (139), (140), , and (147), we have which leads to Since , , and , we obtain from (152) and the boundedness of , , , and that Hence, combining (172) and (175), we get We note that From and (169), we have On the other hand, we observe that By (153), (168), and (176), we have We note that From (178), (180), Lemma 14, and the uniform continuity of , we obtain
Since is bounded, there exists a subsequence of which converges weakly to . From (168) and (176), we have that . From (182) and the uniform continuity of , we have for any . So, from Lemma 16, we have . Similarly to the arguments 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 contraction. So we have . This shows that . Therefore, by Lemma 10, we know that .
Finally, we claim that provided is bounded. Put . Since , we have . By (148) and Lemma 21, we have which converges strongly to some . Since converges weakly to , we have Therefore, we obtain . This completes the proof.

Corollary 31 (i.e., [3, Theorem 4.1]). 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 and be -inverse strongly monotone and -inverse-strongly monotone, respectively, where , . Let be the uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequences and . Assume that is nonempty. Let and be sequences in such that and . Pick any and let be a sequence generated by the following algorithm: where , , , and . Assume that and , and that either (B1) or (B2) holds. Then converges weakly to .

Proof. In Theorem 30, put and for all . Then and . In this case, we obtain from (134) that Thus, the iterative scheme (134) reduces to (95). It is easy to see that all the conditions of Theorem 24 are satisfied. In terms of Theorem 24, we have that converges weakly to an element . Now, put . Since , we have . Taking into account that for all , we conclude from (148) that By Lemma 21, we have that converges strongly to some . Since converges weakly to , we have Therefore, we obtain . This completes the proof.

Corollary 32. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively, where . Let be the uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequences and . Assume that is nonempty. Let is a sequence in , and let be sequences in such that and . Pick any and let be a sequence generated by the following algorithm: where , , and . Assume that either (B1) or (B2) holds and that the following conditions are satisfied:(i) and ;(ii) for some .Then(a) converges weakly to an element ;(b) converges weakly to provided is bounded.

Corollary 33. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively. Let be the uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense for some with sequences and . Assume that is nonempty. Let be a sequence in , and let be sequences in such that and . Pick any and let be a sequence generated by the following algorithm: where , . Assume that either (B1) or (B2) holds and that the following conditions are satisfied:(i) and ;(ii) for some .Then(a) converges weakly to an element ;(b) converges weakly to provided is bounded.

Corollary 34. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively. Let be the uniformly continuous asymptotically -strict pseudocontractive mapping for some with sequence . Assume that is nonempty. Let be a sequence in , and let be sequences in such that and . Pick any and let be a sequence generated by the following algorithm: where . Assume that either (B1) or (B2) holds and that the following conditions are satisfied:(i) and ;(ii) for some .Then(a) converges weakly to an element ;(b) converges weakly to provided is bounded.

Finally, we provide an example to illustrate Corollary 34.

Example 35. Let with inner product and norm which are defined by for all with and . Let . Clearly, is a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient , for instance, putting , where . Then is -Lipschitz continuous with (due to ). Put and for all . Then it is clear that is a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let and be -inverse strongly monotone and -inverse-strongly monotone, respectively, for instance, putting and . Then we can take . Let be the uniformly continuous asymptotically -strict pseudocontractive mapping for some with sequence , for instance, putting . Then and for all (due to the nonexpansivity of ). Thus, we know that . Take , and . Let be a sequence in such that , and let be sequences in such that and . Pick any . In this case, the algorithm (191) reduces to the following algorithm: So it follows that Since , we deduce that there exists an integer such that Therefore, from (194) we obtain that for all This shows that(a) converges to the unique point in ;(b) converges to .

Conflict of Interests

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

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Leading Academic Discipline Project of Shanghai Normal University (DZL707). This research was partially supported by a grant from the NSC (101-2115-M-165 -001) as well.