Journal of Applied Mathematics

Volume 2013 (2013), Article ID 531859, 19 pages

http://dx.doi.org/10.1155/2013/531859

Research Article

Relaxed Viscosity Approximation Methods with Regularization for Constrained Minimization Problems

Lu-Chuan Ceng,¹ Hong-Kun Xu,2,3 and Ching-Feng Wen⁴

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

²Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

³Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

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

Received 13 January 2013; Accepted 26 March 2013

Academic Editor: Luigi Muglia

Copyright © 2013 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 introduce a new relaxed viscosity approximation method with regularization and prove the strong convergence of the method to a common fixed point of finitely many nonexpansive mappings and a strict pseudocontraction that also solves a convex minimization problem and a suitable equilibrium problem.

1. Introduction

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

In particular, if , then is called a nonexpansive mapping. A mapping is called -inverse strongly monotone, if there exists a constant such that

Let be a convex and a continuous Fréchet differentiable functional. Consider the minimization problem (MP) of minimizing over the constraint set

where we assume the existence of minimizers. We denote by the set of minimizers of (3). The gradient-projection algorithm (GPA) generates a sequence determined by the gradient and the metric projection as follows:

or more generally,

where, in both (4) and (5), the initial guess is taken from arbitrarily, the parameters or are positive real numbers. The convergence of algorithms (4) and (5) 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 (4) converges in norm to the unique solution of (3). More generally, if the sequence is chosen to satisfy the property

then the sequence defined by the GPA (5) converges in norm to the unique minimizer of (3). If the gradient is only assumed to be a Lipschitz continuous, then can only be weakly convergent if is infinite dimensional. A counterexample is given by Xu in [1].

Since the Lipschitz continuity of the gradient implies that it is inverse strongly monotone (ism), 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 projectionand an averaged mapping which is again an averaged mapping. This shows that averaged mappings play an important role in the GPA. Very recently, Xu [1] used averaged mappings to study the convergence analysis of the GPA which is an operator-oriented approach.

We observe that the regularization, in particular, the traditional Tikhonov regularization, is usually used to solve ill-posed optimization problems. Consider the following regularized minimization problem:

where is the regularization parameter and again is convex with an -Lipschitz continuous gradient .

The advantage of a regularization method is that it is possible to get strong convergence to the minimum-norm solution of the optimization problem under investigation. The disadvantage is however its implicity, and hence explicit iterative methods seem more attractive. See, for example, [1].

Given a mapping , the classical variational inequality problem (VIP) is to find such that

The solution set of VIP (9) is denoted by . It is well known that if and only if for some . The variational inequality was first discussed by Lions [2] and now is well known. The variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, and equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. See, for example, [3–10] and the references therein.

In this paper, we study the following equilibrium problem (EP) which is to find such that

The solution set of EP (10) is denoted by . We will introduce and consider a relaxed viscosity iterative scheme with regularization for finding a common element of the solution set of the minimization problem (3), the solution set of the equilibrium problem (10), and the common fixed point set of finitely many nonexpansive mappings , and a strictly pseudocontractive mapping in the setting of the infinite-dimensional Hilbert space. We will prove that this iterative scheme converges strongly to a common fixed point of the mappings , which is both a minimizer of MP (3) and an equilibrium point of EP (10).

2. Preliminaries

Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of . We write to indicate that the sequence converges weakly to and to indicate that the sequence converges strongly to . Moreover, we use to denote the weak -limit set of the sequence and to denote the strong -limit set of the sequence ; that is,

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

Some important properties of projections are gathered in the following.

Proposition 1. For given and

(i), for all  ;(ii), for all  ;(iii), for all  , which hence implies that    is nonexpansive and monotone.

Definition 2. A mapping is said to be

(a) nonexpansive if

(b) firmly nonexpansive if is nonexpansive, or equivalently,

alternatively, is firmly nonexpansive if and only if can be expressed as

where is nonexpansive; projections are firmly nonexpansive.

Definition 3. Let be a nonlinear operator with domain and range .

(a) is said to be monotone if

(b) Given a number , is said to be strongly monotone if

(c) Given a number , is said to be -inverse strongly monotone (-ism) if

It can be easily seen that if is nonexpansive, then is monotone. It is also easy to see that a projection is 1-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 maps.

Proposition 5 (see [11]). Let be a given mapping.

(i)  is nonexpansive if and only if the complement    is (1⁄2)-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 [11]). 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, .

It is clear that, in a real Hilbert space , is -strictly pseudocontractive if and only if there holds the following inequality:

This immediately implies that if is a -strictly pseudocontractive mapping, then is -inverse strongly monotone; for further detail, we refer to [12] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings.

Lemma 7 (see [12, Proposition 2.1]). Let be a nonempty closed convex subset of a real Hilbert space and be a mapping.

(i) If is a -strictly pseudocontractive mapping, then satisfies the Lipschitz condition where

(ii) If is a -strictly pseudocontractive mapping, then the mapping is semiclosed at ; that is, if is a sequence in such that weakly and strongly, then .(iii) If is a -(quasi-)strict pseudocontraction, then the fixed point set of is closed and convex so that the projection is well defined.

The following lemma is an immediate consequence of an inner product.

Lemma 8. In a real Hilbert space , there holds the following inequality:

The following elementary result on real sequences is quite well known.

Lemma 9 (see [13]). Let be a sequence of nonnegative real numbers satisfying the property

where and are the real sequences such that

(i);(ii)either or ;(iii) where , for all .

Then, .

Lemma 10 (see [14]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a -strictly pseudocontractive mapping. Let and be two nonnegative real numbers such that . Then,

The following lemma appears implicitly in the paper of Reinermann [15].

Lemma 11 (see [15]). Let be a real Hilbert space. Then, for all and ,

Lemma 12 (see [16]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction such that

(f1) for all ;(f2) is monotone and upper hemicontinuous in the first variable;(f3) is lower semicontinuous and convex in the second variable.

Let be a bifunction such that

(h1) for all ;(h2) is monotone and weakly upper semicontinuous in the first variable;(h3) is convex in the second variable.

Moreover, let one suppose that

(H) for fixed and , there exists a bounded and such that for all .

For and , let be a mapping defined by

called the resolvent of and . Then,

(1);(2) is a singleton;(3) is firmly nonexpansive;(4) and it is closed and convex.

Lemma 13 (see [16]). Let one suppose that (f1)–(f3), (h1)–(h3) and (H) hold. Let , . Then,

Lemma 14 (see [17]). Suppose that the hypotheses of Lemma 12 are satisfied. Let be a sequence in with . Suppose that is a bounded sequence. Then, the following statements are equivalent and true.

(a) if as , each weak cluster point of satisfies the problem:

 that is, .(b) The demiclosedness principle holds in the sense that, if and as , then for all .

3. Main Results

We now propose the following relaxed viscosity iterative scheme with regularization:

for all , where the mapping is a -contraction; the mapping is a -strict pseudocontraction;   is a nonexpansive mapping for each ;   satisfies the Lipschitz condition (10) with ;    are two bifunctions satisfying the hypotheses of Lemma 12;   is a sequence in with ;    are sequences in with ;   are sequences in with , for all ;    are sequences in and , for all ; is a sequence in with and .

Before stating and proving the main convergence results, we first establish the following lemmas.

Lemma 15. Let one suppose that . Then, the sequences , , for all , and are bounded.

Proof. First of all, we can show as in [18] that is nonexpansive for , and is nonexpansive for all and . We observe that if , then

For all, from to , by induction, one proves that

Thus, we obtain that for every ,

For simplicity, put and for every . Then, and for every . Taking into consideration that and for , we have

Similarly, we get . Thus, from (34) we have

Since for all , utilizing Lemma 10, we derive from (35)

By induction, we get

This implies that is bounded and so are , , and for each . It is clear that both and are also bounded. Since , is also bounded.

Lemma 16. Let one suppose that . Moreover, let one suppose that the following hold:

(H1) and ;(H2) or ;(H3) or for each ;(H4) or ;(H5) or ;(H6) or .

Then, , that is, is asymptotically regular.

Proof. Taking into account , we may assume, without loss of generality, that for some . First, we write , for all , where ). It follows that for all

Since for all , utilizing Lemma 10, we have

Next, we estimate . Observe that for every

and similarly,

Also, from (30), we have

Simple calculations show that

Then, passing to the norm we get from (40) that

where , for all for some . Furthermore, by the definition of one obtains that, for all

In the case of , we have

Substituting (46) in all (45) type one obtains for

This together with (44) implies that

By Lemma 13, we know that

where . So, substituting (49) in (48) we obtain

where is a minorant for and , for all for some . This together with (38)-(39), implies that

where , for all for some .

Further, we observe that

Simple calculations show that

Then, passing to the norm, we get from (51)

where , for all for some . By hypotheses (H1)–(H6) and Lemma 9, from , we obtain the claim.

Lemma 17. Let one suppose that . Let one suppose that is asymptotically regular. Then, and as .

Proof. We recall that, by the firm nonexpansivity of , a standard calculation (see [17]) shows that if , then

Let . Then by Lemma 11, we have from (33)–(34) the following

Since for all , utilizing Lemma 10, we have

Taking into account , we may assume that for some . So, we deduce that

Since , and as , we conclude from the boundedness of , and that as . This together with , implies that

Furthermore, from (33), (55), and (56), we have

which hence implies that

Since and as , we deduce from the boundedness of , , and that

Remark 18. By the last lemma we have and ; that is, the sets of strong/weak cluster points of and coincide.

Of course, if , as , for all index , the assumptions of Lemma 16 are enough to assure that

In the next lemma, we examine the case in which at least one sequence is a null sequence.

Lemma 19. Let one suppose that . Let one suppose that (H1) holds. Moreover, for an index , , and the following hold:

(H7) for all ,

(H8) there exists a constant such that for all . Then,

Proof. We start by (54). Dividing both the terms by we have

So, by (H8) we have

Therefore, utilizing Lemma 9, from (H1), (H7), and the asymptotical regularity of (due to Lemma 16), we deduce that

Lemma 20. Let one suppose that . Let one suppose that (H1)–(H6) hold. Then,

Proof. Let . Then, by Lemma 11 we have

So, we obtain

Since , , , and , from the boundedness of , , and it follows that , and hence

Moreover, from the firm nonexpansiveness of we obtain

and so

Thus, we have

which implies that

Since , , and , from the boundedness of , , , and , it follows that . Observe that

and hence

Thus, .

Lemma 21. Let one suppose that . Let one suppose that for each . Moreover, suppose that (H1)–(H6) are satisfied. Then, for each .

Proof. First of all, observe that

By Lemmas 16 and 20, we know that and as . Hence, utilizing Lemma 7(i), we have

which together with implies that . Taking into account , we have

Let us show that for each , one has as . Let . When , by Lemma 11, we have from (33)-(34) the following:

So, we have

Since , , , and , it is known that is a null sequence.

Let . Then, one has

and so, after iterations,

Again, we obtain that

Since , , for each , and , it is known that

Obviously, for , we have .

To conclude, we have that

from which . Thus, by induction for all since it is enough to observe that

Remark 22. As an example, we consider and the following sequences:

(a), for all ;(b), , for all ;(c), for all .

Then, they satisfy the hypotheses on the parameter sequences in Lemma 21.

Lemma 23. Let one suppose that and for all as . Suppose there exists such that as . Let be the largest index such that as . Suppose that

(i) as ;(ii) if and , then as ;(iii) if , then lies in .

Moreover, suppose that (H1), (H7), and (H8) hold. Then, for each .

Proof. First of all, we note that if (H7) holds, then also (H2)–(H6) are satisfied. So is asymptotically regular. Let be as in the hypotheses. As in Lemma 21, for every index such that (which leads to ), one has as .

For all the other indexes , we can prove that as in a similar manner. By the following relation (due to (86)):

we immediately obtain that

By Lemma 19 or by hypothesis (ii) of the sequences, we have

So, the thesis follows.

Remark 24. Let us consider and the following sequences:

(a), , , for all ;(b), , for all ;(c), , , for all .

It is easy to see that all hypotheses (i)–(iii), (H1), (H7), and (H8) of Lemma 23 are satisfied.

Remark 25. Under the hypotheses of Lemma 23, analogously to Lemma 21, one can see that

Corollary 26. Let one suppose that the hypotheses of either Lemma 21 or Lemma 23 are satisfied. Then, , and .

Proof. By Remark 18, we have and . Observe that

By Lemmas 17 and 21, and as , and hence

So, we get and .

Let . Since , by Lemma 21 and Lemma 7(ii) (demiclosedness principle), we have for all index , that is, . Taking into consideration that is -strictly pseudocontractive, by Lemma 7(i), we get

which together with (by Lemma 17) and (by (82)) implies that

Utilizing Lemma 7(ii) (demiclosedness principle), we have . Furthermore, by Lemmas 14 and 17, we know that . Finally, by similar argument as in [18], we can show that , and as a result .

Theorem 27. Let one suppose that . Let , , , be sequences in such that for all index . Moreover, Let one suppose that (H1)–(H6) hold. Then, the sequences , , and , explicitly defined by scheme (30), all converge strongly to the unique solution of the following variational inequality:

Proof. Since the mapping is a -contraction, it has a unique fixed point ; it is the unique solution of (99). Since (H1)–(H6) hold, the sequence is asymptotically regular (by Lemma 16). In terms of Lemma 17, and as . Moreover, utilizing Lemmas 8 and 10, we have from (33)-(34) the following:

Now, let be a subsequence of such that

By the boundedness of , we may assume, without loss of generality, that . According to Corollary 26, we know that , and hence . Taking into consideration that we obtain from (101) that

Since and , we deduce that and . In terms of Lemma 9 we derive as .

In a similar way, we can derive the following result.

Theorem 28. Let one suppose that . Let , , , be sequences in such that for all as . Suppose that there exists for which as . Let the largest index for which . Moreover, let one suppose that (H1), (H7), and (H8) hold, and

(i) as ;(ii) if and , then as ;(iii)if , then lies in .

Then, the sequences , , and explicitly defined by scheme (30) all converge strongly to the unique solution of the following variational inequality:

Remark 29. According to the above argument processes for Theorems 27 and 28, we can readily see that if in scheme (30), the iterative step is replaced by the iterative one , then Theorems 27 and 28 remain valid.

Remark 30. Theorems 27 and 28 improve, extend, supplement, and develop [17, Theorems 3.12 and 3.13] and [1, Theorems 5.2 and 6.1] in the following aspects:

(a)the multistep iterative scheme (30) of [17] is extended to develop our relaxed viscosity iterative scheme (30) with regularization for MP (3), EP (10), and strict pseudocontraction by virtue of Xu iterative schemes in [1];(b)the argument techniques in Theorems 27 and 28 are very different from the ones in [17, Theorems 3.12 and 3.13] and the ones in [1, Theorems 5.2 and 6.1] because we use the properties of strict pseudocontractive mappings and maximal monotone mappings (see, e.g., Lemmas 7 and 10);(c)compared with the proof of Theorems 5.2 and 6.1 in [1], the proof of Theorems 27 and 28 shows that via the argument of , for all (see Lemma 20 and its proof);(d)the problem of finding an element of in Theorems 27 and 28 is more general than the one of finding an element of in [17, Theorems 3.12 and 3.13] and the one of finding an element of in [1, Theorems 5.2 and 6.1].

4. Applications

For a given nonlinear mapping , we consider again the variational inequality problem (VIP) of finding such that

Recall that if is a point in , then the following relation holds:

from which we have the following relation:

An operator is said to be an -inverse strongly monotone operator if there exists a constant such that

As an example, we recall that the -inverse strongly monotone operators are firmly nonexpansive mappings if and that every -inverse strongly monotone operator is also a ) Lipschitz continuous (see [19]). We observe that, if is -inverse strongly monotone, the mapping is nonexpansive for all since they are compositions of nonexpansive mappings (see [19, page 419]).

Let us consider a finite number of nonexpansive self-mappings on and be a finite number of -inverse strongly monotone operators. Let be a -strict pseudocontraction on with fixed points. Let us consider the following mixed problem of finding such that

We denote by (SVI) the set of solutions of the above system. This problem is equivalent to finding a common fixed point of , , . The following results are then consequences of Theorems 27 and 28.

Theorem 31. Let one suppose that .  , and . Let , , , be sequences in such that for all index . Moreover, Let one suppose that (H1)–(H6) hold. Then the sequences , , and explicitly defined by the following scheme:

all converge strongly to the unique solution of the following variational inequality:

Theorem 32. Let one suppose that . and . Let , , , be sequences in and for all as . Suppose that there exists such that as . Let be the largest index for which . Moreover, let one suppose that (H1), (H7), and (H8) hold, and

(i) as ;(ii)if and , then as ;(iii)if , then lies in .

Then, the sequences , , and explicitly defined by scheme (109) all converge strongly to the unique solution of the following variational inequality:

Remark 33. If we choose in system (108), we obtain a system of hierarchical fixed point problems introduced by Moudafi and Maingé [20, 21].

Further, utilizing Theorems 27 and 28, we again give the following strong convergence theorems for finding a common element of the solution set of MP (3), the solution set of EP (10), and the common fixed point set of a finite family of nonexpansive mappings .

Theorem 34. Let one suppose that . Let , , , be sequences in such that for all index . Moreover, Let one suppose that there hold (H1)–(H6) with , for all . Then, the sequences , , and generated explicitly by

all converge strongly to the unique solution of the following variational inequality:

Proof. In Theorems 27, put the identity mapping and , for all . Then, is a -strictly pseudocontractive mapping with . Hence, we deduce that , , for all , and

Thus, the conditions in Theorem 27 are all satisfied. and from which we obtain the desired result.

Theorem 35. Let one suppose that . Let , , , be sequences in such that for all as . Suppose that there exists for which as . Let be the largest index for which . Moreover, let one suppose that there hold (H1), (H7), and (H8) with , for all , and

(i) as ;(ii)if and , then as ;(iii)if , then lies in .

Then the sequences , , and generated explicitly by (112) all converge strongly to the unique solution of the following variational inequality:

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Leading Academic Discipline Project of Shanghai Normal University (DZL707). This research was partially supported by the Grant from the NSC 101-2115-M-037-001.

References

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.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, France, 1969.

L. C. Zeng, “Iterative algorithms for finding approximate solutions for general strongly nonlinear variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 187, no. 2, pp. 352–360, 1994.

J. C. Yao, “Variational inequalities with generalized monotone operators,” Mathematics of Operations Research, vol. 19, no. 3, pp. 691–705, 1994.

L. C. Zeng, S. Schaible, and J. C. Yao, “Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities,” Journal of Optimization Theory and Applications, vol. 124, no. 3, pp. 725–738, 2005.

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 S. Huang, “Modified extragradient methods for strict pseudo-contractions and monotone mappings,” Taiwanese Journal of Mathematics, vol. 13, no. 4, pp. 1197–1211, 2009.

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.

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.

L.-C. Zeng and J.-C. Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,” Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1293–1303, 2006.

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.

G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007.

H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society. Second Series, vol. 66, no. 1, pp. 240–256, 2002.

Y. Yao, Y.-C. Liou, and S. M. Kang, “Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3472–3480, 2010.

J. Reinermann, “Über Fixpunkte kontrahierender Abbildungen und schwach konvergente Toeplitz-Verfahren,” Archiv der Mathematik, vol. 20, pp. 59–64, 1969.

F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem,” Fixed Point Theory and Applications, vol. 2010, Article ID 383740, 19 pages, 2010.

G. Marino, L. Muglia, and Y. Yao, “Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 4, pp. 1787–1798, 2012.

L. C. Ceng, A. Petruşel, and J. C. Yao, “Relaxed extragradient methods with regularization for general system of variational inequalities with constraints of split feasibility and fixed point problems,” Abstract and Applied Analysis, vol. 2013, Article ID 891232, 25 pages, 2013.

W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.

A. Moudafi and P.-E. Maingé, “Towards viscosity approximations of hierarchical fixed-point problems,” Fixed Point Theory and Applications, vol. 2006, Article ID 95453, 10 pages, 2006.

P.-E. Maingé and A. Moudafi, “Strong convergence of an iterative method for hierarchical fixed-point problems,” Pacific Journal of Optimization, vol. 3, no. 3, pp. 529–538, 2007.