Abstract

We introduce and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: finitely many generalized mixed equilibrium problems, finitely many variational inequalities, the general system of variational inequalities and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm by hybrid shrinking projection method for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.

1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space and let be the metric projection of onto . Let be a nonlinear mapping on . We denote by the set of fixed points of and by the set of all real numbers. 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. A mapping is called strongly positive on if there exists a constant such that

Let be a nonlinear mapping on . We consider the following variational inequality problem (VIP): find a point such that The solution set of VIP (3) is denoted by .

Let be a real-valued function, let be a nonlinear mapping, and let be a bifunction. Peng and Yao [1] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that We denote the set of solutions of GMEP (4) by . The GMEP (4) 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. It covers problems considered in [2–5].

It is assumed 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.

Given a positive number , let be the solution set of the auxiliary mixed equilibrium problem; that is, for each ,

Let be two mappings. Consider the following general system of variational inequalities (GSVI) [6] of finding such that where and are two constants. In 2008, Ceng et al. [6] transformed the GSVI (8) into a fixed point problem in the following way.

Proposition CWY (see [6]). For given is a solution of the GSVI (8) if and only if is a fixed point of the mapping defined by where .

In particular, if the mapping is -inverse-strongly monotone for , then the mapping is nonexpansive provided for . We denote by the fixed point set of the mapping .

Let , . Given the nonexpansive self-mappings on , for each , the mappings are defined by

The is called the -mapping generated by and . Note that the nonexpansivity of implies the one of   . In 2012, combining the hybrid steepest-descent method in [7] and viscosity approximation method, Ceng et al. [8] proposed and analyzed the following hybrid iterative algorithm for finding a common element of the solution set of GMEP (4) and the fixed point set of finitely many nonexpansive mappings .

Theorem  CGY (see [8, Theorem 3.1]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction satisfying assumptions (A1)–(A4) and let be a lower semicontinuous and convex function with restriction (B1) or (B2). Let the mapping be -inverse-strongly monotone, and let be a finite family of nonexpansive mappings on such that . Let be a -Lipschitzian and -strongly monotone operator with positive constants and an -Lipschitzian mapping with constant . Let and , where . Suppose and are two sequences in , is a sequence in , and is a sequence in with . For every , let be the -mapping generated by and . Given arbitrarily, suppose the sequences and are generated iteratively by where the sequences , and and the finite family of sequences satisfy the following conditions:(i) and ;(ii) ;(iii) and ;(iv) for . Then both and converge strongly to , which is the unique solution in to the VIP

Let be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing over the constraint set We denote by the set of minimizers of CMP (13).

Next, recall some concepts. Let be a nonempty subset of a normed space . A mapping is called uniformly Lipschitzian if there exists a constant such that Recently, Kim and Xu [9] introduced the concept of asymptotically -strict pseudocontractive mappings in a Hilbert space as below.

Definition 1. 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
It is important to note that every asymptotically -strict pseudocontractive mapping with sequence is a uniformly -Lipschitzian mapping with . Subsequently, Sahu et al. [10] considered the concept of asymptotically -strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.

Definition 2. 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 there holds the relation

In 2009, Sahu et al. [10] first established one weak convergence theorem for the following Mann-type iterative scheme: where , , and , and then obtained another strong convergence theorem for the following hybrid CQ iterative scheme: where , , and . Subsequently, the above iterative schemes 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; see, for example, [11–13].

Motivated and inspired by the above facts, we first introduce and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the CMP (13) with constraints of several problems: finitely many GMEPs, finitely many VIPs, the GSVI (8), and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. The iterative algorithm is based on shrinking projection method, Korpelevich's extragradient method, hybrid steepest-descent method in [7], viscosity approximation method, averaged mapping approach to the GPA in [14], and strongly positive bounded linear operator technique. On the other hand, we also propose another iterative algorithm by hybrid shrinking projection method for finding a fixed point of infinitely many nonexpansive mappings with the same constraints. We derive its strong convergence under mild assumptions. The results obtained in this paper improve and extend the corresponding results announced by many others.

2. Preliminaries

Throughout this paper, we assume that is 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,

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.

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 3. For given and ,(i) , ;(ii) , ;(iii) , .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 if , then is a nonexpansive mapping from to .

Definition 4. 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 cocoercive) operators have been applied widely in solving practical problems in various fields.

Definition 5. 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 mappings.

Proposition 6 (see [15]). 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 7 (see [15]). 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, then The notation denotes the set of all fixed points of the mapping ; that is, .

Proposition 8 (see [3]). 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.

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 monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 3(i)) implies

Lemma 11 (see [16, 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 . Here is the identity operator of .
Let be an infinite family of nonexpansive mappings on and let be a sequence of nonnegative numbers in . For any , define a mapping on as follows: Such a mapping is called the -mapping generated by and .

Lemma 12 (see [17, 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, where is defined as in (34).

Lemma 13 (see [17, 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, .

The following lemma can be easily proven, and, therefore, we omit the proof.

Lemma 14. Let be a -strongly positive bounded linear operator with constant . Then, for , That is, is strongly monotone with constant .
Let be a nonempty closed convex subset of a real Hilbert space . We introduce some notations. Let be a number in and let . Associating with a nonexpansive mapping , we define the mapping by where is an operator such that, for some positive constants , is -Lipschitzian and -strongly monotone on ; that is, satisfies the following conditions: for all .

Lemma 15 (see [18, Lemma 3.1]). is a contraction provided ; that is, where .

Lemma 16 ([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).

Recall that a set-valued mapping is called monotone if, for all , and imply A set-valued mapping is called maximal monotone if is monotone and for each , where is the identity mapping of . We denote by the graph of . It is known that a monotone mapping is maximal if and only if, for for every implies . Let be a monotone, -Lipschitz-continuous mapping and let be the normal cone to at ; that is, Define Then, is maximal monotone and if and only if .

Lemma 17 ([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 18 ([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 19 (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 20 ([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.

Remark 21. Lemmas 19 and 20 give some basic properties of an asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence .

Lemma 22 (see [19]). 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 .

Lemma 23. 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

3. Convex Minimization Problems with Constraints

In this section, we will introduce and analyze one iterative algorithm by hybrid shrinking projection method for finding a solution of the CMP (13) with constraints of several problems: finitely many GMEPs, finitely many VIPs, GSVI (8), and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. This iterative algorithm is based on shrinking projection method, Korpelevich's extragradient method, hybrid steepest-descent method in [7], viscosity approximation method, averaged mapping approach to the GPA in [14], and strongly positive bounded linear operator technique.

Theorem 24. Let be a nonempty closed convex subset of a real Hilbert space . Let be two integers. 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, where . Let , and be -inverse-strongly monotone, -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 -strongly positive bounded linear operator with . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that is nonempty and bounded and that either (B1) or (B2) holds. Let , for all , and let , be sequences in . Pick any and set , . Let be a sequence generated by the following algorithm: where here is nonexpansive; for each , , and . Suppose that the following conditions are satisfied:(i) for each , ;(ii) , and , where , , and ;(iii) . Then one has the following:(I) converges strongly as to ;(II) converges strongly as to provided and , which is the unique solution in to the VIP Equivalently, .

Proof . Since is -Lipschitzian, it follows that is -ism. By Proposition 6(ii) we know that, for is -ism. So by Proposition 6(iii) we deduce that is -averaged. Now since the projection is -averaged, it is easy to see from Proposition 7(iv) that the composite is -averaged for . Hence we obtain that, for each , is -averaged for each . Therefore, we can write where is nonexpansive and for each . It is clear that As and , we may assume, without loss of generality, that and for all . Since is a -strongly positive bounded linear operator on , we know that Taking into account that for all , we have that is, is positive. It follows that Put for all and and for all , , and , where is the identity mapping on . Then we have that and .
We divide the rest of 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 16 we know that is convex for every .
First of all, let us show that for all . Suppose that for some . Take arbitrarily. From (46) and Proposition 8(iii), we have Similarly, we have Combining (56) and (57), we have Since , is -inverse-strongly monotone for , and for , we deduce that, for any , Utilizing Lemma 15, from (46), (52), (58), and (59), we obtain that which hence yields By Lemma 23(b), we deduce from (46) and (61) that So, from (46) and (62) we get where and . Hence . This implies that for all . Therefore, is well defined.
Step 2. We prove that , and 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 3(ii), we obtain which implies It follows from that and hence From (67) and , we have Since and , we have which immediately leads to Also, utilizing Lemmas 9 and 23(b) we obtain from (46), (58), (59), and (62) that and hence So, it follows that Since , , and , it follows from (69) and the boundedness of , and that Note that Hence, it follows from (75) and that Note that Thus, we deduce from (71) and (77) that Since and , we have which, together with (79), yields
Step 3. We prove that , , , , and as .
Indeed, from (57), (59), , and it follows that Next let us show that For , we find from (46) that By (56), (82), and (84), we obtain which immediately yields Since , and and are bounded sequences, it follows from (77) that By Proposition 8(iii) and (46), we have which implies that From (82) and (89), we have which leads to Since , , and and are bounded sequences, it follows from (77) and (87) that Hence we obtain from (92) that That is, (83) holds.
Next we show that , . As a matter of fact, observe that Combining (57), (82), and (94), we have which leads to Since , , and and are bounded sequences, it follows from (77) that By Proposition 3(iii) and Lemma 23(a), we obtain which implies
Combining (57), (82), and (99), we have which yields Since and , , and are bounded, from (77) and (97) we get From (102) we get Taking into account that , we conclude from (83) and (103) that
On the other hand, for simplicity, we write , , and for all . Then
We now show that ; that is, . As a matter of fact, for , it follows from (58), (59), and (82) that which immediately yields Since and and are bounded, from (77) we get Also, in terms of the firm nonexpansivity of and the -inverse strong monotonicity of for , we obtain from , , and (59) that Thus, we have Consequently, from (58), (106), and (110) it follows that which hence leads to Since and , and are bounded sequences, we conclude from (77) and (108) that Furthermore, from (58), (106), and (111) it follows that which hence yields Since and , and are bounded sequences, we conclude from (77) and (108) that Note that Hence from (114) and (117) we get Observe that Hence, from (75), (104), and (119) we have It is clear that where for each . Hence we have From the boundedness of , ( ), and (due to (121)), it follows that In addition, from (67) and (77), we have We note that From (81), (125), and Lemma 17, we obtain In the meantime, we note that From (81), (127), and the uniform continuity of , we have
Step 4. We prove that as .
Indeed, since is bounded, there exists a subsequence which converges weakly to some . From (77), (83), (104), (92), and (102) we have that , , , , and , where and . Since is uniformly continuous, by (129) we get for any . Hence from Lemma 19, we obtain . In the meantime, utilizing Lemma 11, we deduce from , , (119), and (124) that and . 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 (102) and since is Lipschitz continuous, we obtain that . From , , , and (102), 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 (92), we have as . Furthermore, by the monotonicity of , we obtain . Then, by (A4) we obtain Utilizing (A1), (A4), and (139), we obtain and hence Letting , we have, for each , This implies that and hence . Consequently, . This shows that . From (64) and Lemma 22 we infer that as .
Finally, assume additionally that and . It is clear that So, we know that is -strongly monotone with constant . In the meantime, it is easy to see that is -Lipschitzian with constant . Thus, there exists a unique solution in to the VIP Equivalently, . Furthermore, from (58), (59), and (82) we get which hence yields Since , , , and , are bounded, we infer from (146) that which, together with Minty's Lemma [4], implies that This shows that is a solution in to the VIP (144). Utilizing the uniqueness of solutions in to the VIP (144), we get . This completes the proof.

Corollary 25. 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, -inverse-strongly monotone, and -inverse-strongly monotone, respectively, for 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 -strongly positive bounded linear operator with . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that is nonempty and bounded and that either (B1) or (B2) holds. Let for all , and let , be sequences in . Pick any and set . Let be a sequence generated by the following algorithm: where (here is nonexpansive; for each ), , and . Suppose that the following conditions are satisfied:(i) for each , ( );(ii) , and for and ;(iii) .
Then one has the following:(I) converges strongly as to ;(II) converges strongly as to provided and , which is the unique solution in to the VIP Equivalently, .

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, -inverse-strongly monotone, and -inverse-strongly monotone, respectively, for . 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 -strongly positive bounded linear operator with . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that is nonempty and bounded and that either (B1) or (B2) holds. Let , for all , and let , be sequences in . Pick any and set , . Let be a sequence generated by the following algorithm: where (here is nonexpansive; for each ), , and . Suppose that the following conditions are satisfied:(i) for each , ;(ii) , , and for ;(iii) .
Then one has the following:(I) converges strongly as to ;(II) converges strongly as to provided and , which is the unique solution in to the VIP Equivalently, .

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, -inverse-strongly monotone, and -inverse-strongly monotone, respectively, for . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping for some with sequence such that . Let be a -strongly positive bounded linear operator with . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that is nonempty and bounded and that either (B1) or (B2) holds. Let , for all , and let be sequences in . Pick any and set , . Let be a sequence generated by the following algorithm: where (here is nonexpansive; for each ), , and . Suppose that the following conditions are satisfied:(i) for each , ( );(ii) , and for ;(iii) .
Then one has the following:(I) converges strongly as to ;(II) converges strongly as to provided and , which is the unique solution in to the VIP Equivalently, .

4. Fixed Point Problems with Constraints

In this section, we will introduce and analyze another implicit iterative algorithm for solving the fixed point problem of infinitely many nonexpansive mappings with constraints of several problems: finitely many GMEPs, finitely many VIPs, the GSVI (8), and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under mild assumptions. This iterative algorithm is based on shrinking projection method, Korpelevich's extragradient method, hybrid steepest-descent method in [7], viscosity approximation method, -mapping approach to fixed points of infinitely many nonexpansive mappings, and strongly positive bounded linear operator technique.

Theorem 28. 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, -inverse-strongly monotone, and -inverse-strongly monotone, respectively, where , , and . Let be a sequence of nonexpansive mappings on and let be a sequence in for some . 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 -strongly positive bounded linear operator with . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that is nonempty and bounded and that either (B1) or (B2) holds. Let , for all , and let , and be sequences in . Pick any and set , . Let be a sequence generated by the following algorithm: where is the -mapping defined by (34), , and . Suppose that the following conditions are satisfied:(i) , and , where , , and ;(ii) and .
Then one has the following:(I) converges strongly to ;(II) converges strongly to provided and , which is the unique solution in to the VIP Equivalently, .

Proof. First of all, let us show that the sequence is well defined. As and , we may assume, without loss of generality, that and for all . Utilizing the arguments similar to those in the proof of Theorem 24, we get Put for all and and for all , , and , where is the identity mapping on . Then we have that and .
We divide the rest of 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 16 we know that is convex for every .
First of all, let us show that for all . Suppose that for some . Take arbitrarily. Utilizing the arguments similar to those in the proof of Theorem 24 we obtain that So, from (155) and (165) we get where and . Hence . This implies that for all . Therefore, is well defined.
Step 2. We prove that , , and as .
Indeed, let . From and , we obtain This implies that is bounded and hence , and are also bounded. Utilizing the arguments similar to those of (67), (75), (77), and (81) in the proof of Theorem 24 we obtain that
Step 3. We prove that , , , , and as .
Indeed, from (162), (164), , and , it follows that Utilizing the arguments similar to those of (83), (92), (102), (104), (119), (121), and (129) in the proof of Theorem 24 we obtain that In addition, note that So, from and [20, Remark 3.2] it follows that
Step 4. We prove that as .
Indeed, since is bounded, there exists a subsequence which converges weakly to some . From (169), (172), (175), (173), and (174) we have that , and , where and . Since is uniformly continuous, by (178) we get for any . Hence, from Lemma 19, we obtain . In the meantime, utilizing Lemma 11, we deduce from (176) and (180) that and (due to Lemma 13). Hence we get . Repeating the same arguments as in the proof of Theorem 24 we conclude that and . Consequently, . This shows that . From (167) and Lemma 22 we infer that as .
Finally, assume additionally that and . It is clear that So, we know that is -strongly monotone with constant . In the meantime, it is easy to see that is -Lipschitzian with constant . Thus, there exists a unique solution in to the VIP Equivalently, . Furthermore, from (163), (164), and (171) we get which hence yields Since , , , and are bounded, we infer from (94) that which, together with Minty's Lemma, implies that This shows that is a solution in to the VIP (182). Utilizing the uniqueness of solutions in to the VIP (182), we get . This completes the proof.

Corollary 29. Let be a nonempty closed convex subset of a real Hilbert space . 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, -inverse-strongly monotone, and -inverse-strongly monotone, respectively, for and . Let be a sequence of nonexpansive mappings on and let be a sequence in for some . 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 -strongly positive bounded linear operator with . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that is nonempty and bounded and that either (B1) or (B2) holds. Let , for all , and let , and be sequences in . Pick any and set , . Let be a sequence generated by the following algorithm: where is the -mapping defined by (34), , and . Suppose that the following conditions are satisfied:(i) , and for and ;(ii) and . Then one has the following:(I) converges strongly to ;(II) converges strongly to provided and , which is the unique solution in to the VIP Equivalently, .

Corollary 30. Let be a nonempty closed convex subset of a real Hilbert space . 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, -inverse-strongly monotone, and -inverse-strongly monotone, respectively, for . Let be a sequence of nonexpansive mappings on and let be a sequence in for some . 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 -strongly positive bounded linear operator with . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that is nonempty and bounded and that either (B1) or (B2) holds. Let , for all , and let , and be sequences in . Pick any and set , . Let be a sequence generated by the following algorithm: where is the -mapping defined by (34), , and . Suppose that the following conditions are satisfied:(i) , and for ;(ii) and . Then one has the following:(I) converges strongly to ;(II) converges strongly to provided and , which is the unique solution in to the VIP Equivalently, .

Corollary 31. Let be a nonempty closed convex subset of a real Hilbert space . 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, -inverse-strongly monotone, and -inverse-strongly monotone, respectively, for . Let be a sequence of nonexpansive mappings on and let be a sequence in for some . Let be a uniformly continuous asymptotically -strict pseudocontractive mapping for some with sequence such that . Let be a -strongly positive bounded linear operator with . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that is nonempty and bounded and that either (B1) or (B2) holds. Let for all , and let , and be sequences in . Pick any and set . Let be a sequence generated by the following algorithm: where is the -mapping defined by (34), , and . Suppose that the following conditions are satisfied:(i) , and for ;(ii) and . Then one has the following:(I) converges strongly to ;(II) converges strongly to provided and , which is the unique solution in to the VIP Equivalently, .

Remark 32. Let be -inverse-strongly monotone and let be -inverse-strongly monotone for . Let be a -contraction with , and 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. In [11], Guu et al. introduced and analyzed a hybrid viscosity CQ iterative algorithm for finding a point : where for ; ; is a sequence in ; and , and are three sequences in such that for all . The authors of [11] proved that under suitable conditions converges strongly to ; see [11, Theorem 3.1] for more details.
Theorem 28 extends, improves, supplements, and develops [11, Theorem 3.1] in the following aspects.(i)The problem of finding a point in Theorem 28 is very different from the problem of finding a point in [11, Theorem 3.1]. There is no doubt that our problem of finding a point is more general and more subtle than the problem of finding a point in [11, Theorem 3.1].(ii) The iterative scheme in [11, Theorem 3.1] is extended to develop the iterative scheme in Theorem 28 by virtue of Cai and Bu iterative algorithm in [21, Theorem 3.1] and Ceng et al. iterative one in [8, Theorem 3.1]. The iterative scheme in Theorem 28 is more advantageous and more flexible than the iterative scheme in [11, Theorem 3.1] because it involves solving four problems: the GSVI (8), finitely many GMEPs, finitely many VIPs, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings on .(iii)The iterative scheme in Theorem 28 is very different from the iterative scheme in [11, Theorem 3.1] because the iterative scheme in our theorem (Theorem 28) involves hybrid steepest-descent method in [7], strongly positive bounded linear operator technique, finitely many GMEPs, finitely many VIPs, and infinitely many nonexpansive mappings. The proof in [11, Theorem 3.1] makes use of Proposition CWY and the properties of asymptotically strict pseudocontractive mapping in the intermediate sense (see Lemmas 17–20). However, the proof of Theorem 28 depends on not only Proposition CWY and Lemmas 17–20 but also Proposition 8, the properties of strongly positive bounded linear operator , and the ones of the -mapping and -mapping (see Lemmas 12, 13, and 15) because there are the mapping , infinitely many nonexpansive mappings , -Lipschitzian and -strongly monotone operator , and strongly positive bounded linear operator appearing in the iterative scheme of our theorem (Theorem 28).(iv)The proof of Theorem 28 combines Cai and Bu convergence analysis for their iterative algorithm to solve finitely many GMEPs, finitely many VIPs, and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense (see [21, Theorem 3.1]); the convergence analysis for the -mapping approach to fixed points of infinitely many nonexpansive mappings and strongly positive bounded linear operator technique; and Ceng, Guu, and Yao convergence analysis for hybrid iterative method (see [11, Theorem 3.1]).

Remark 33. Theorem 28 also extends, improves, supplements, and develops Ceng et al. [8, Theorem 3.1] in the following aspects.(i)The problem of finding a point in Theorem 28 is very different from the problem of finding a point in Ceng et al. [8, Theorem 3.1]. Here our problem of finding a point is put forth after one GMEP; finitely many nonexpansive mappings in their problem are replaced by finitely many GMEPs and infinitely many nonexpansive mappings, respectively; and the GSVI (8), finitely many VIPs, and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense are added to their problem.(ii)The iterative scheme in [8, Theorem 3.1] is extended to develop the iterative scheme in our theorem (Theorem 28) by virtue of Korpelevich's extragradient method [22], shrinking projection method, Mann iterative method, and strongly positive bounded linear operator technique. The iterative scheme in Theorem 28 is put forth after and in [8, Theorem 3.1] are replaced by and , respectively.(iii)The iterative scheme in Theorem 28 is very different from the iterative scheme in [8, Theorem 3.1] because the iterative scheme in Theorem 28 involves Korpelevich's extragradient method [22], shrinking projection method, Mann iterative method, and strongly positive bounded linear operator technique. The proof of [8, Theorem 3.1] makes use of Proposition 8. However, the proof of Theorem 28 depends on not only Proposition 8 but also Proposition CWY, the properties of strongly positive bounded linear operator, and the ones of asymptotically strict pseudocontractive mapping in the intermediate sense (see Lemmas 17–20) because there are the SGEP (8), finitely many GMEPs, asymptotically strict pseudocontractive mapping in the intermediate sense, and the strongly positive bounded linear operator appearing in the iterative scheme of our theorem (Theorem 28).(iv)The proof of Theorem 28 involves the convergence analysis for Korpelevich's extragradient method to solve the SGEP (8), finitely many GMEPs, and finitely many VIPs; the convergence analysis for the -mapping approach to fixed points of infinitely many nonexpansive mappings and strongly positive bounded linear operator technique; and Ceng, Guu, and Yao convergence analysis for viscosity approximation method and hybrid steepest-descent method (see [23, Theorem 4.2] and [8, Theorem 3.1]).

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), the Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and the Ph.D. Program Foundation of Ministry of Education of China (20123127110002). This work was supported partly by the National Science Council of the Republic of China. This research was partially supported by a grant from NSC.