Abstract and Applied Analysis

Volume 2014 (2014), Article ID 587865, 26 pages

http://dx.doi.org/10.1155/2014/587865

Research Article

Solving Generalized Mixed Equilibria, Variational Inequalities, and Constrained Convex Minimization

A. E. Al-Mazrooei,¹ A. Latif,¹ and J. C. Yao1,2

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 14 October 2013; Accepted 18 November 2013; Published 8 January 2014

Academic Editor: Chi-Ming Chen

Copyright © 2014 A. E. Al-Mazrooei 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 propose implicit and explicit iterative algorithms 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 a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings in a real Hilbert space. We prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets under very mild conditions.

1. Introduction and Problems Formulation

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

In particular, if , then is called a nonexpansive mapping [1], and 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.

Let be a nonlinear mapping on . We consider the following variational inequality problem (VIP): find a point such that

The solution set of VIP (5) is denoted by .

The VIP (5) was first discussed by Lions [2] and is now well known. The VIP (5) has many potential applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and so on; see, for example, [3–5] and the references therein.

In 1976, Korpelevich [6] proposed an iterative algorithm for solving the VIP (5) in Euclidean space :

with , a given number which is known as the extragradient method. The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many researchers. See, for example, [7–16] and the references therein. In particular, motivated by the idea of Korpelevich’s extragradient method [6], Nadezhkina and Takahashi [17] introduced an extragradient iterative scheme:

where is a monotone, -Lipschitz continuous mapping, is a nonexpansive mapping, for some , and for some . They proved the weak convergence of to an element of .

Let be a real-valued function, let be a nonlinear mapping, and let be a bifunction. In 2008, Peng and Yao [18] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that

We denote the set of solutions of GMEP (8) by . The GMEP (8) 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. See, for example, [19, 20]. Some special cases of GMEP (8) are as follows.

If , then GMEP (8) reduces to the generalized equilibrium problem (GEP) which is to find such that

It is introduced and studied by S. Takahashi and W. Takahashi [21]. The set of solutions of GEP is denoted by .

If , then GMEP (8) reduces to the mixed equilibrium problem (MEP) which is to find such that

It is considered and studied in [22]. The set of solutions of MEP is denoted by .

If and , then GMEP (8) reduces to the equilibrium problem (EP) which is to find such that

It is considered and studied in [23]. The set of solutions of EP is denoted by .

Throughout this paper, it is assumed as in [18] that is a bifunction satisfying conditions (A1)–(A4) and is a lower semicontinuous and convex function with restriction (B1) or (B2), where

 (A1), for all ; (A2) is monotone; that is, for any ; (A3) is upper hemicontinuous; that is, for each ,

 (A4) is convex and lower semicontinuous for each ; (B1)for each and , there exists a bounded subset and such that, for any ,

 (B2) is a bounded set.

Next we list some known results for the MEP as follows.

Proposition 1 (see [22]). 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 conditions hold:

(i)for each , is nonempty and single-valued; (ii) is firmly nonexpansive; that is, for any ,

(iii);(iv) is closed and convex;(v), for all and .

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

The is called the -mapping generated by and . Note that the nonexpansivity of implies the nonexpansivity of .

In 2012, combining the hybrid steepest-descent method in [24] and hybrid viscosity approximation method in [25], Ceng et al. [20] proposed and analyzed the following hybrid iterative method for finding a common element of the set of solutions of GMEP (8) and the set of fixed points of a finite family of nonexpansive mappings .

Theorem CGY (see [20, 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 constants and a -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 all .

Then both and converge strongly to , where is a unique solution of the variational inequality problem ():

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 (19). It is well known that the gradient-projection algorithm (GPA) generates a sequence determined by the gradient and the metric projection :

or more generally,

where, in both (20) and (21), the initial guess is taken from arbitrarily and the parameters or are positive real numbers. The convergence of algorithms (20) and (21) depends on the behavior of the gradient . As a matter of fact, it is known that, if is -strongly monotone and -Lipschitz continuous, then, for , the operator

is a contraction. Hence, the sequence defined by the GPA (20) converges in norm to the unique solution of CMP (19). More generally, if the sequence is chosen to satisfy the property

then the sequence defined by the GPA (21) converges in norm to the unique minimizer of CMP (19). If the gradient is only assumed to be Lipschitz continuous, then can only be weakly convergent if is infinite dimensional (a counterexample is given in Section 5 of Xu [26]).

Since the Lipschitz continuity of the gradient implies that it is actually -inverse strongly monotone (ism) [27], 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 [26] used averaged mappings to study the convergence analysis of the GPA, which is hence an operator-oriented approach.

In 2011, combining the hybrid steepest-descent method in [24], viscosity approximation method, and averaged mapping approach to the GPA in [26], Ceng et al. [28] introduced and analyzed the following implicit and explicit iterative algorithms:

where is -Lipschitzian mapping with constant and is a -Lipschitzian and -strongly monotone operator with constants . Assume that , , for each , for each , with and , and . The authors proved that the net defined by (24) converges strongly to some , which is a unique solution of the variational inequality problem (VIP):

Furthermore, utilizing control conditions (i) , (ii) , and (iii) either or , the authors also proved that the sequence generated by (25) converges strongly to some , which is a unique solution of the VIP (26).

Motivated and inspired by the above facts, in this paper we introduce implicit and explicit iterative algorithms for finding a common element of the set of solutions of the CMP (19) for a convex functional with -Lipschitz continuous gradient , the set of solutions of a finite family of GMEPs, and the set of solutions of a finite family of VIPs for inverse strong monotone mappings in a real Hilbert space. Under very mild control conditions, we prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets. Our iterative algorithms are based on Korpelevich’s extragradient method, hybrid steepest-descent method in [24], viscosity approximation method, and averaged mapping approach to the GPA in [26]. 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 with inner product and norm 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 projection from onto is the mapping which assigns to each point the unique point satisfying the property

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

Proposition 2. 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 3. A mapping is said to be

(a)nonexpansive [1] 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 4. 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 5 (see [29]). 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 6 (see [29]). 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, .

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

Lemma 7. Let be a real inner product space. Then there holds the following inequality:

Lemma 8. Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 2(i)) implies

Lemma 9 (see [30, Demiclosedness principle]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive self-mapping on with . 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 10 (see [31]). Let be a sequence of nonnegative numbers satisfying the conditions

where and are sequences of real numbers such that

(i) and , or, equivalently,

(ii), or . Then .

Lemma 11 (see [32]). Let and be bounded sequences in a Banach space and let be a sequence in with

Suppose that for each and

Then .

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

Lemma 12. Let be an -Lipschitzian mapping with constant , and let be a -Lipschitzian and -strongly monotone operator with positive constants . 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 13 (see [31, Lemma 3.1]). is a contraction provided ; that is,

where .

Remark 14. (i) Since is -Lipschitzian and -strongly monotone on , we get . Hence, whenever , we have

which implies

So, .

(ii) In Lemma 13, put and . Then we know that , and

Finally, recall that 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, , for all . Define

It is known that in this case is maximal monotone, and if and only if ; see [33].

3. Implicit Iterative Algorithm and Its Convergence Criteria

We now state and prove the first main result of this paper.

Theorem 15. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let and 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 and . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that and that either (B1) or (B2) holds. Let be a sequence generated by

where (here is nonexpansive and for each ). Assume that the following conditions hold:

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

Then converges strongly as () to a point , which is a unique solution of the VIP:

Equivalently, .

Proof. First of all, let us show that the sequence is well defined. Indeed, since is -Lipschitzian, it follows that is -ism; see [34]. By Proposition 5(ii) we know that, for is -ism. So by Proposition 5(iii) we deduce that is -averaged. Now since the projection is -averaged, it is easy to see from Proposition 6(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

Put

for all and ,

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

Consider the following mapping on defined by

where for each . By Proposition 1(ii) and Lemma 13 we obtain from (29) 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 and . Hence by Lemma 12 we know that

That is, is strongly monotone for . Moreover, it is clear that is Lipschitz continuous. So the VIP (50) has only one solution. Below we use to denote the unique solution of the VIP (50).

Now, let us show that is bounded. In fact, take arbitrarily. Then from (29) and Proposition 1(ii) we have

Similarly, we have

Combining (59) and (60), we have

Since

where . It is clear that for each . Thus, utilizing Lemma 13 and the nonexpansivity of , we obtain from (61) that

This implies that . Hence is bounded. So, according to (59) and (61) we know that , and are bounded.

Next let us show that , , and as .

Indeed, from (29) it follows that, for all and ,

Thus, utilizing Lemma 7, from (49) and (64) we have

which implies that

Since and , for all and , from we conclude immediately that

for all and .

Furthermore, by Proposition 1(ii) we obtain that for each

which implies that

Also, by Proposition 2(iii), we obtain that for each

which implies

Thus, utilizing Lemma 7, from (49), (69), and (71) we have

It immediately follows that

Since and , for all and , from (67) and we deduce that

for all and . Hence we get

So, taking into account that , we have

Thus, from (77) and we have

Now we show that as . In fact, from the nonexpansivity of , we have

By (77) and (78), we get

From (78) it is easy to see that

Observe that

where for each . Hence we have

From the boundedness of   ) and   due to (78)), it follows that

Further, we show that . Indeed, since is bounded, there exists a subsequence of which converges weakly to some . Note that   due to (75)). Hence . Since is closed and convex, is weakly closed. So, we have . From (74)-(75), we have that , , , and , where and . First, we prove that . Let

where . Let . Since and , we have

On the other hand, from and , we have

and hence

Therefore we have

From (74) and since is Lipschitz continuous, we obtain that . From , for all and (74), 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 (74), we have as . Furthermore, by the monotonicity of , we obtain . Then, by (A4) we obtain

Utilizing (A1), (A4), and (94), we obtain

and hence

Letting , we have, for each ,

This implies that and hence . Further, let us show that . As a matter of fact, from (84), , and Lemma 9, we conclude that

So, . Therefore, . This shows that .

Finally, let us show that as , where is the unique solution of the VIP (50). Indeed, we note that, for with ,

By (61) and Lemma 13, we obtain that

Hence it follows that

which hence leads to

In particular, we have

Since , it follows from (103) that as .

Now we show that solves the VIP (50). Since , we have

It follows that, for each ,

since is monotone (i.e., , for all . This is due to the nonexpansivity of ). Since as , by replacing in (105) with and letting , we get

That is, is a solution of VIP (50).

Finally we show that the sequence converges strongly to . To this end, let be another subsequence of such that . By the same arguments as above, we have . Moreover, it follows from (106) that

Interchanging and , we obtain

Utilizing Lemma 12 and adding the two inequalities (107) and (108), we have

Hence . Therefore we conclude that as . Taking into account the uniqueness of solutions of VIP (50), we have . The VIP (50) can be rewritten as

By Proposition 2(i), this is equivalent to the fixed point equation

This completes the proof.

Corollary 16. 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, for . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that and that either (B1) or (B2) holds. Let be a sequence generated by

where (here is nonexpansive and for each ). Assume that the following conditions hold:

(i) for each ,   ;(ii) for ;(iii).

Then converges strongly as () to a point , which is a unique solution of the VIP:

Corollary 17. 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 -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that and that either (B1) or (B2) holds. Let be a sequence generated by

where (here is nonexpansive and for each ). Assume that the following conditions hold:

(i) for each ,   ;(ii);(iii).

Then converges strongly as () to a point , which is a unique solution of the VIP:

4. Explicit Iterative Algorithm and Its Convergence Criteria

We next state and prove the second main result of this paper.

Theorem 18. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let and 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 and . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that and that either (B1) or (B2) holds. For arbitrarily given , let be a sequence generated by

where (here is nonexpansive and for each ). Assume that the following conditions hold:

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

Then converges strongly as () to a point , which is a unique solution of VIP (50).

Proof. First of all, repeating the same arguments as in Theorem 15, we can write

where is nonexpansive and for each . It is clear that

Put

for all and ,

for all and , and , where is the identity mapping on . Then we have that and . In addition, taking into consideration conditions (i) and (ii), we may assume, without loss of generality, that , for all .

We divide the remainder of the proof into several steps.

Step   1. Let us show that , for all and . Indeed, take arbitrarily. Repeating the same arguments as those of (59)–(61) in the proof of Theorem 15, we obtain

Taking into account conditions (i) and (ii), we may assume, without loss of generality, that , for all . Then from (121), , and Lemma 13, we have

By induction, we have

Hence is bounded. According to (121), , , , , and are also bounded.

Step   2. Let us show that as . To this end, define

Observe that, from the definition of ,

Thus, it follows that

On the other hand, since is -ism, is nonexpansive for . So, it follows that, for any given ,

This together with the boundedness of implies that is bounded. Also, observe that

where for some . So, by (128), we have that

Note that

where for some . Also, utilizing Proposition 1(ii), (v) we deduce that

where is a constant such that, for each ,

Combining (126)–(131), we get

Thus, it follows from (133) and conditions (i)–(iv) that

Hence by Lemma 11 we have

Consequently,

and by (129)–(131),

Step   3. Let us show that and , for all and .

Indeed, since

we have

that is,

So, from , , and condition (ii), it follows that

Also, from (29) it follows that, for all and ,

Furthermore, utilizing Lemma 7, we deduce from (116) that

From (142)-(143), it follows that

and so

Since and , for all and , by (136), (141), and (145) we conclude immediately that

for all and .

Step   4. Let us show that , , and as .

Indeed, by Proposition 1(ii) we obtain that for each

which implies that

Also, by Proposition 2(iii), we obtain that for each

which implies

Thus, from (143), (148), and (150), we have

that is,

So, from , (136), (141), and (146) we immediately get

for all and . Note that

Thus, from (153) we have

It is easy to see that as

Also, observe that

Hence we have from (141)

Step   5. Let us show that , where is the same as in Theorem 15; that is, is a unique solution of VIP (50). To show this inequality, we choose a subsequence of such that

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we may assume that . From Step 4, we have that , , , and , where and . Since by Step 4, by the same arguments as in the proof of Theorem 15, we get . Since , it follows that

Step   6. Let us show that , where is the same as in Theorem 15; that is, is a unique solution of VIP (50). From (116), we know that

Applying Lemmas 7 and 13 and noticing that and , for all , we have

This implies that

where , , and

From condition (i) and Step 5, it is easy to see that and . Hence, by Lemma 10, we conclude that as . This completes the proof.

Corollary 19. 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, for . Let be a -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that and that either (B1) or (B2) holds. For arbitrarily given , let be a sequence generated by

where (here is nonexpansive and for each ). Assume that the following conditions hold:

(i) for each , and   ;(ii) and ;(iii) and for ;(iv) and .

Then converges strongly as to a point , which is a unique solution of VIP (113).

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 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 -Lipschitzian and -strongly monotone operator with positive constants . Let be an -Lipschitzian mapping with constant . Let and , where . Assume that and that either (B1) or (B2) holds. For arbitrarily given , let be a sequence generated by

where (here is nonexpansive and for each ). Assume that the following conditions hold:

(i) for each , and   ;(ii) and ;(iii) and ;(iv) and .

Then converges strongly as to a point , which is a unique solution of VIP (115).

Conflict of Interests

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

Acknowledgment

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant no. HiCi/15-130-1433. The authors, therefore, acknowledge the technical and financial support of KAU.

References

F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.

J. L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites non Linéaires, Dunod, Paris, 1969.

R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, USA, 1984.

J. T. Oden, Quantitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, NJ, USA, 1986.

E. Zeidler, Nonlinear Functional Analysis and its Applications, Springer, New York, NY, USA, 1985.

G. M. Korpelevich, “The extragradient method for finding saddle points and other problems,” Matecon, vol. 12, pp. 747–756, 1976.

L.-C. Ceng, Q. H. Ansari, M. M. Wong, and J.-C. Yao, “Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems,” Fixed Point Theory, vol. 13, no. 2, pp. 403–422, 2012.

L.-C. Ceng and J.-C. Yao, “An extragradient-like approximation method for variational inequality problems and fixed point problems,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 205–215, 2007.

L.-C. Ceng and J.-C. Yao, “A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1922–1937, 2010.

L.-C. Ceng and A. Petruşel, “Relaxed extragradient-like method for general system of generalized mixed equilibria and fixed point problem,” Taiwanese Journal of Mathematics, vol. 16, no. 2, pp. 445–478, 2012.

L.-C. Ceng, C.-y. Wang, and J.-C. Yao, “Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities,” Mathematical Methods of Operations Research, vol. 67, no. 3, pp. 375–390, 2008.

L.-C. Ceng, Q. H. Ansari, and S. Schaible, “Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems,” Journal of Global Optimization, vol. 53, no. 1, pp. 69–96, 2012.

L.-C. Ceng, Q. H. Ansari, M. M. Wong, and J.-C. Yao, “Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems,” Fixed Point Theory, vol. 13, no. 2, pp. 403–422, 2012.

L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “An extragradient method for solving split feasibility and fixed point problems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 633–642, 2012.

L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2116–2125, 2012.

L. C. Ceng, M. Teboulle, and J. C. Yao, “Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 19–31, 2010.

N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 128, no. 1, pp. 191–201, 2006.

J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 12, no. 6, pp. 1401–1432, 2008.

L. C. Ceng, H.-Y. Hu, and M. M. Wong, “Strong and weak convergence theorems for generalized mixed equilibrium problem with perturbation and fixed pointed problem of infinitely many nonexpansive mappings,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 1341–1367, 2011.

L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems,” Fixed Point Theory and Applications, vol. 2012, article 92, p. 19, 2012.

S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008.

L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.

L. C. Ceng, A. Petruşel, and J. C. Yao, “Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Journal of Optimization Theory and Applications, vol. 143, no. 1, pp. 37–58, 2009.

I. Yamada, “The hybrid steepest-descent method for the variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Batnariu, Y. Censor, and S. Reich, Eds., pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.

V. Colao, G. Marino, and H.-K. Xu, “An iterative method for finding common solutions of equilibrium and fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 340–352, 2008.

H.-K. Xu, “Averaged mappings and the gradient-projection algorithm,” Journal of Optimization Theory and Applications, vol. 150, no. 2, pp. 360–378, 2011.

J.-B. Baillon and G. Haddad, “Quelques propriétés des opérateurs angle-bornés et -cycliquement monotones,” Israel Journal of Mathematics, vol. 26, no. 2, pp. 137–150, 1977.

L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Some iterative methods for finding fixed points and for solving constrained convex minimization problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 16, pp. 5286–5302, 2011.

C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004.

K. Geobel and W. A. Kirk, Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, UK, 1990.

H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003.

T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.

R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976.

J.-B. Baillon and G. Haddad, “Quelques propriétés des opérateurs angle-bornés et -cycliquement monotones,” Israel Journal of Mathematics, vol. 26, no. 2, pp. 137–150, 1977.