Abstract and Applied Analysis

Volume 2014 (2014), Article ID 209372, 22 pages

http://dx.doi.org/10.1155/2014/209372

Research Article

Iterative Schemes for Convex Minimization Problems with Constraints

Lu-Chuan Ceng,¹ Cheng-Wen Liao,² Chin-Tzong Pang,³ and Ching-Feng Wen⁴

¹Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China

²Department of Food and Beverage Management, Vanung University, Chung-Li 320061, Taiwan

³Department of Information Management and Innovation Center for Big Data and Digital Convergence, Yuan Ze University, Chung-Li 32003, Taiwan

⁴Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 17 February 2014; Accepted 12 April 2014; Published 8 May 2014

Academic Editor: Xie-ping Ding

Copyright © 2014 Lu-Chuan Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We first introduce and analyze one implicit iterative algorithm for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: the generalized mixed equilibrium problem, the system of generalized equilibrium problems, and finitely many variational inclusions 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 implicit iterative algorithm 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 strongly positive on if there exists a constant such that

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.

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, a nonlinear mapping, and a bifunction. The generalized mixed equilibrium problem (GMEP) introduced in [1] is to find such that

We denote the set of solutions of GMEP (4) by .

We assume as in [1] that is a bifunction satisfying conditions (H1)–(H4) and is a lower semicontinuous and convex function with restriction (H5), where

(H1) for all ;(H2) is monotone, that is, for any ;(H3) is upper-hemicontinuous, that is, for each ,

(H4) is convex and lower semicontinuous for each ;(H5)for each and , there exist a bounded subset and such that, for any ,

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

In particular, whenever , for  all is rewritten as .

Let be two bifunctions and two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP) [2]: find such that

where and are two constants.

In 2010, Ceng and Yao [2] transformed the SGEP into a fixed point problem in the following way.

Proposition CY (see [2]). Let be two bifunctions satisfying conditions (H1)–(H4) and let be -inverse-strongly monotone for . Let for . Then, is a solution of SGEP (8) if and only if is a fixed point of the mapping defined by where . Here, we denote the fixed point set of by .

Let be an infinite family of nonexpansive mappings on and a sequence of nonnegative numbers in . For any , define a mapping on as follows:

Such a mapping is called the -mapping generated by and .

Let be a contraction and a strongly positive bounded linear operator on . Assume that is a lower semicontinuous and convex functional, that satisfy conditions (H1)–(H4), and that are inverse-strongly monotone. Let the mapping be defined as in Proposition CY. Very recently, Ceng et al. [3] introduced the following hybrid extragradient-like iterative scheme:

for finding a common solution of GMEP (4), SGEP (8), and the fixed point problem of an infinite family of nonexpansive mappings on , where , , ,  , and are given. The authors proved the strong convergence of the sequence to a point under some suitable conditions. This point also solves the following optimization problem:

where is the potential function of .

Let be a single-valued mapping of into and a multivalued mapping with . Consider the following variational inclusion: find a point such that

We denote by the solution set of the variational inclusion (11). In 1998, Huang [4] studied problem (11) in the case where is maximal monotone and is strongly monotone and Lipschitz continuous with .

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 (12).

Very recently, Ceng and Al-Homidan [5] introduced an implicit iterative algorithm for finding a common solution of the CMP (12), finitely many GMEPs and finitely many variational inclusions, and derived its strong convergence under appropriate conditions.

Algorithm CA (see [5, Theorem  18]). 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 (H1)–(H4) and 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 -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Let be a sequence generated by

where (here is nonexpansive, for each ), and the following conditions hold:

(i) for each , ();(ii),  for  all  ;(iii),  for  all  .

Motivated and inspired by the above facts, we first introduce and analyze one implicit iterative algorithm for finding a solution of the CMP (12) with constraints of several problems: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. The iterative algorithm is based on Korpelevich's extragradient method, hybrid steepest-descent method in [6], viscosity approximation method, averaged mapping approach to the GPA in [7], and strongly positive bounded linear operator technique. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints. We derive its strong convergence under mild assumptions.

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 1. 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. If , then it is easy to see that is a nonexpansive mapping from to .

Definition 2. 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.

Definition 3. 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 4 (see [8]). 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 5 (see [8]). 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, .

By using the technique in [9], we can readily obtain the following elementary result.

Proposition 6 (see [3, Lemma 1 and Proposition 1]). Let be a nonempty closed convex subset of a real Hilbert space and let be a lower semicontinuous and convex function. Let be a bifunction satisfying the conditions (H1)–(H4). Assume that

(i) is strongly convex with constant and the function is weakly upper semicontinuous for each ;(ii) for each and , there exists a bounded subset and such that, for any ,

Then the following hold:

(a) for each , ;(b) is single-valued;(c) is nonexpansive if is Lipschitz continuous with constant and

where for ;

(d) for all and

(e);(f) is closed and convex.

Remark 7. In Proposition 4, whenever is a bifunction satisfying the conditions (H1)–(H4) and ,  for  all  , we have for any ,

( is firmly nonexpansive) and

In this case, is rewritten as . If, in addition, , then is rewritten as ; see [2, Lemma 2.1] for more details.

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

Lemma 8. Let be a real inner product space. Then there holds the following inequality:

Lemma 9. Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 1(i)) implies

Lemma 10 (see [10, 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 .

Lemma 11 (see [11, 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 (9).

Lemma 12 (see [11, 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 proved and, therefore, we omit the proof.

Lemma 13. 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 conditions

for all .

Lemma 14 (see [12, Lemma  3.1]). is a contraction provided ; that is,

where .

We observe that since is -Lipschitzian and -strongly monotone on , we get . Hence, whenever , we have . Also in Lemma 14, put and . Then we know that and .

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

It is well known that is maximal monotone and if and only if .

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

From Huang [4] (see also [13]), there holds the following property for the resolvent operator .

Lemma 15. is single-valued and firmly nonexpansive; that is,

Consequently, is nonexpansive and monotone.

Lemma 16 (see [14]). Let be a maximal monotone mapping with . Then for any given , is a solution of problem (11) if and only if satisfies

Lemma 17 (see [13]). 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 18 (see [14]). Let be a maximal monotone mapping with and a monotone, continuous, and single-valued mapping. Then for each . In this case, is maximal monotone.

3. Convex Minimization Problems with Constraints

In this section, we will introduce and analyze one implicit iterative algorithm for finding a solution of the CMP (12) with constraints of several problems: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We po prove strong convergence theorem for the iterative algorithm under suitable conditions.

Theorem 19. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be a convex functional with -Lipschitz continuous gradient . Let be three bifunctions from to satisfying (H1)–(H4) and a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and let be -inverse strongly monotone, -inverse strongly monotone, and -inverse-strongly monotone, respectively, for and . 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 where is defined as in Proposition CY. Let be a sequence in and a sequence in such that . Let be the sequence generated by

where (here is nonexpansive, for each ). Suppose that the following conditions hold.

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;(ii)for each , there exist a bounded subset and such that, for any ,

(iii) for each ,   ;(iv),   and ,  ;(v).

Assume that is firmly nonexpansive. Then converges strongly as to a point , which is a unique solution in to the VIP:

Proof. First of all, let us show that the sequence is well defined. Indeed, since is -Lipschitzian, it follows that is -ism. By Proposition 4(ii) we know that, for is -ism. So by Proposition 4(iii) we deduce that is -averaged. Now since the projection is -averaged, it is easy to see from Proposition 5(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 , where is the identity mapping on . Then we have .

Consider the following mapping on defined by

where for each . Since is -inverse-strongly monotone with for , we deduce that. for any ,

By the nonexpansivity of and Lemma 14 we obtain from (41), (48), and (51) that for all

Since , is a contraction. Therefore, by the Banach contraction principle, has a unique fixed point , which uniquely solves the fixed point equation

This shows that the sequence is defined well.

Note that

That is, is -strongly monotone for . Moreover, it is clear that is Lipschitz continuous. So the VIP (43) has only one solution in . Below we use to denote the unique solution of the VIP (43).

Now, let us show that is bounded. In fact, take arbitrarily. Then from the nonexpansivity of and we have

Since , and is -inverse strongly monotone, where ,  , by Lemma 15 we deduce that for each

Combining (55) and (56), we have

Since is -inverse-strongly monotone for , and for , we deduce that, for any ,

Thus, utilizing Lemmas 8 and 14, from (41), (43), (55), (56), and (58) we have

which implies that

Hence is bounded. So, according to (55) and (57) we know that , , and are bounded.

Next let us show that , , and as . Indeed, combining (55) and (59), we obtain

which immediately yields

From and the boundedness of we conclude that

Furthermore, from the firm nonexpansivity of , we have

which leads to

From (59) and (65), we get

which hence implies that

Since and and are bounded sequences, it follows from (63) that

Next we show that ,  . As a matter of fact, combining (55), (56), and (59), we have

which leads to

Since ,  , and is bounded, it follows that

Also, by Lemma 15, we obtain that for each

which yields

Thus, utilizing Lemma 8, from (41), (55), (56), (59), and (73) we have

which leads to

Since ,   and and are bounded sequences, it follows from (71) that

From (76) we get

Taking into account that , we conclude from (68) and (77) 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 (56), (57), and (59) that

which immediately implies that

Since , ,   and is bounded, it follows that

Also, in terms of the firm nonexpansivity of and the -inverse strong monotonicity of for , we obtain from ,  , and (58) that

Thus, we have

Consequently, from (57), (80), and (84) it follows that

which hence leads to

Since and , and are bounded sequences, we conclude from (82) that

Consequently, from (57), (80), and (85) it follows that

which hence yields

Since and , and are bounded sequences, we conclude from (82) that

Note that

Hence from (88) and (91) we get

Also, it is clear from (41) that

So, it follows from that

Observe that

Hence, from (78), (93), and (95) we have

It is easy to see that

where for each . Hence we have

From the boundedness of    and (due to (97)), it follows that

Furthermore, we show that . Indeed, since is bounded, there exists a subsequence   of   which converges weakly to some . Note that (due to (68)). Hence . Since is closed and convex, is weakly closed. So, we have . From (68), (76), and (78) we have that , and , where . First, we prove that . However, the argument to show that lies in is quite standard by using maximality and hence is omitted. Next, we show that . In fact, from , we know that

From (H2) it follows that

Replacing by , we have

Put for all and . Then, from (103) we have

Since as , we deduce from the Lipschitz continuity of and that and as . Furthermore, from the monotonicity of , we have . So, from (H4), the weakly lower semicontinuity of , and , we have

From (H1), (H4), and (105) we also have

and hence

Letting , we have, for each ,

This implies that . In addition, let us show that . As a matter of fact, from (93), (100), , and Lemma 10, we deduce that and . Hence we get . Therefore, . This shows that .

Finally, we prove that converges strongly as () to , which is the unique solution in to the VIP (43). In fact, we note that, for with ,

By (48), (57), and Lemma 14, we obtain that

which hence leads to

In particular, we have

Since and , it follows from (112) that as .

Now we show that solves the VIP (43). As a matter of fact, from (55) and (59) we obtain that, for any ,

which immediately implies that

Since and , we get

By Minty's lemma, is a solution in to the VIP (43). In terms of the uniqueness of solutions of VIP (43), we deduce that and as . So, every weak convergence subsequence of converges strongly to the unique solution of VIP (43). Therefore, converges strongly to the unique solution of VIP (43). This completes the proof.

Corollary 20. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be three bifunctions from to satisfying (H1)–(H4) and let be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse-strongly monotone, respectively, for and . 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 where is defined as in Proposition CY. Let be a sequence in and a sequence in such that . Let be the sequence generated by

where   here is nonexpansive, for each . Suppose that the following conditions hold.

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each .(ii)for each , there exist a bounded subset and such that for any ,

(iii) for each ,   .(iv),   and ,  .(v).

Assume that is firmly nonexpansive. Then converges strongly as    to a point , which is a unique solution in to the VIP:

Corollary 21. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be three bifunctions from to satisfying (H1)–(H4) and a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse-strongly monotone, respectively, for . 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 where is defined as in Proposition CY. Let be a sequence in and be a sequence in such that . Let be the sequence generated by

where   here is nonexpansive, for each . Suppose that the following conditions hold

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;(ii)for each , there exist a bounded subset and such that for any ,

(iii) for each ,   ;(iv),   and ;(v).

Assume that is firmly nonexpansive. Then converges strongly as () to a point , which is a unique solution in to the VIP:

We next provide one example to illustrate Corollary 21.

Example 22. Let with inner product and norm which are defined by

for all with and . Let and ,  for  all  . Clearly, is a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient ; for instance, putting and , we obtain that and with Lipschitz constant . Put and . Then are three bifunctions from to satisfying (H1)–(H4) and let be a lower semicontinuous and convex functional. Let be a maximal monotone mapping, for instance, putting

where ,  for  all  . It is known that is maximal monotone and if and only if . Put , , , Then and are -inverse strongly monotone with , -inverse strongly monotone with and -inverse-strongly monotone with , respectively, for . Also, put , , , and , where is the identity mapping of . Then is a -strongly positive bounded linear operator with . It is easy to see that where is defined as in Proposition CY. Let , , and with ; that is, . In this case, for any given , the iterative scheme (119) is equivalent to the following one:

where and for each with ; that is, for each .

Next, taking into account , we get , which leads to

Hence, we have ,  for  all  ; that is,

which immediately implies that . Thus,

Also, note that

So, we obtain

On the other hand, we have

and hence

which together with (124) implies that

It is clear that if for each , then, for , we deduce from (132) that

Therefore, for all . There is no doubt that converges to the unique element in , which solves the VIP (121).

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: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under mild assumptions.

Theorem 23. Let be a nonempty closed convex subset of a real Hilbert space . Let be an integer. Let be three bifunctions from to satisfying (H1)–(H4) and a lower semicontinuous and convex functional. Let be a maximal monotone mapping and 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 a sequence in for some . 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 where is defined as in Proposition CY. Let be a sequence in and let and be sequences in . Let be the sequence generated by

where is the -mapping defined by (9). Suppose that the following conditions hold:

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;(ii)for each , there exist a bounded subset and such that for any ,

(iii) and ;(iv) and ,  ;(v).

Assume that is firmly nonexpansive. Then converges strongly to a point , which is a unique solution in to the VIP:

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 19, we get

Put

for all , and , where is the identity mapping on . Then we have .

Consider the following mapping on defined by

Since is -inverse-strongly monotone with for , repeating the same arguments as in the proof of Theorem 19 we deduce that is a nonexpansive mapping on . Utilizing the arguments similar to those in the proof of Theorem 19, we deduce that, for any ,

Since , is a contraction. Therefore, by the Banach contraction principle, has a unique fixed point , which uniquely solves the fixed point equation

This shows that the sequence is defined well.

It is easy to see the VIP (43) has only one solution in . Below we use to denote the unique solution of the VIP (43).

Now, let us show that is bounded. In fact, take arbitrarily. Repeating the same arguments as in the proof of Theorem 19 we obtain

From (145) we conclude that

Hence is bounded. So, according to (142) and (143) we know that , , and are bounded.

Repeating the arguments similar to those of (68), (76), (78), (93), and (97) in the proof of Theorem 19 we obtain that , , , and as . In addition, note that

So, from and [15, Remark 3.2] it follows that

Further, we show that . Indeed, since is bounded, there exists a subsequence of which converges weakly to some . Note that . Hence . Since is closed and convex, is weakly closed. So, we have . On the other hand, it is easy to see that , , and , where . Repeating the same arguments as in the proof of Theorem 19, we obtain that . Next let us show that . As a matter of fact, from , , and Lemma 10, we deduce that and (due to Lemma 12). Hence we get . Therefore, . This shows that .

Finally, we prove that converges strongly to , which is the unique solution in to the VIP (136). In fact, we note that, for with ,

Utilizing the arguments similar to those in the proof of Theorem 19, we obtain that

which hence leads to

In particular, we have

Since and , it follows from (152) that as .

Now we show that solves the VIP (136). As a matter of fact, from (142) and (145) we obtain that, for any ,

which immediately implies that

Since and , we get

By Minty's lemma, is a solution in to the VIP (136). In terms of the uniqueness of solutions of VIP (136), we deduce that and as . So, every weak convergence subsequence of converges strongly to the unique solution of VIP (136). Therefore, converges strongly to the unique solution of VIP (136). This completes the proof.

Corollary 24. Let be a nonempty closed convex subset of a real Hilbert space . Let be three bifunctions from to satisfying (H1)–(H4) and let be a lower semicontinuous and convex functional. Let be a maximal monotone mapping and 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 -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 where is defined as in Proposition CY. Let be a sequence in and and sequences in . Let be the sequence generated by

where is the -mapping defined by (9). Suppose that the following conditions hold:

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;(ii)for each , there exist a bounded subset and such that, for any ,

(iii) and ;(iv) and ,  ;(v).

Assume that is firmly nonexpansive. Then converges strongly to a point , which is a unique solution in to the VIP:

Corollary 25. Let be a nonempty closed convex subset of a real Hilbert space . Let be three bifunctions from to satisfying (H1)–(H4) and a lower semicontinuous and convex functional. Let be a maximal monotone mapping and let and be -inverse strongly monotone, -inverse strongly monotone, and -inverse strongly monotone, respectively, for . Let be a sequence of nonexpansive mappings on and a sequence in for some . 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 where is defined as in Proposition CY. Let be a sequence in and and sequences in . Let be the sequence generated by

where is the -mapping defined by (9). Suppose that the following conditions hold:

(i) is strongly convex with constant and its derivative is Lipschitz continuous with constant such that the function is weakly upper semicontinuous for each ;(ii)for each , there exist a bounded subset and such that, for any ,

(iii) and ;(iv),   and ;(v).

Assume that is firmly nonexpansive. Then converges strongly to a point , which is a unique solution in to the VIP:

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 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.

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