#### Abstract

The purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of fixed points of quasi-nonexpansive mappings and the solution of split feasibility problems (SFP) and systems of equilibrium problems (SEP) in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge weakly to a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems under mild conditions. Our main result improves and extends the recent ones announced by Ceng et al. (2012) and many others.

#### 1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space . A mapping is said to be *nonexpansive* if for all . Denote the set of fixed points of by . On the other hand, a mapping is said to be *quasi*-*nonexpansive* if and for all and . If is nonexpansive and the set of fixed points of is nonempty, then is quasi-nonexpansive. Fixed point iterations process for nonexpansive mappings and quasi-nonexpansive mappings in Banach spaces including Mann and Ishikawa iterations process have been studied extensively by many authors to solve the nonlinear operator equations (see [1–4]).

Let be a bifunction of into , where is the set of real numbers. The equilibrium problem for is to find such that The set of solutions of (1) is denoted by . Numerous problems in physics, optimization, and economics reduce to find a solution of (1) in Hilbert spaces; see, for instance, Blum and Oettli [5], Flam and Antipin [6], and Moudafi [7]. Moreover, Flam and Antipin [6] introduced an iterative scheme of finding the best approximation to the solution of equilibrium problem, when is nonempty, and proved a strong convergence theorem (see also in [8–11]). Let be two-monotone bifunction and is a constant. Recently, Moudafi [12] considered the following of a system of equilibrium problem, denoting the set of solution of SEP by , for finding such that He also proved the weak convergence theorem of this problem (some related work can be found in [13, 14]).

The split feasibility problem (SFP) in Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction was first introduced by Censor and Elfving [15] (see, e.g. [16, 17]). It has been found that the SFP can also be used to model the intensity-modulated radiation therapy (see [18, 19]). In this work, the SFP is formulated as finding a point with the property where and are the nonempty closed convex subsets of the infinite-dimensional real Hilbert spaces and , respectively, and (i.e., is a bounded linear operator from to ). Very recently, there are related works which we can find in [16, 18, 20–26] and the references therein.

A special case of the SFP is called the convex constrained linear inverse problem (see [27]), that is, the problem to finding an element such that In fact, it has been extensively investigated in the literature using the projected Landweber iterative method [27, 28]. Throughout this paper, we assume that the solution set of the is nonempty.

Motivated and inspired by the regularization method and extragradient method due to Ceng et al. [29], we introduce and analyze an extragradient method with regularization for finding a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems (SFP) and systems of equilibrium problems (SEP) in Hilbert spaces. Our results represent the improvement, extension, and development of the corresponding results in [14, 29].

#### 2. Preliminaries

Let be a real Hilbert space with inner product and norm , and let be a closed convex subset of . We write to indicate that the sequence converges weakly to and to indicate that the sequence converges strongly to . For every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto .

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

Proposition 1 (see [29]). *For given and : *(i);
(ii).

*Definition 2 (see [30, 31]). *Let be a nonlinear operator with domain and range , and let and be given constants. The operator is called(a)*monotone* if
(b)*-strongly monotone* if
(c)*-inverse strongly monotone* (*ism*) if

We can easily see that if is nonexpansive, then is monotone. It is also easy to see that a projection is a 1-*ism*.

*Definition 3 (see [29]). *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 (9) holds, we say that is . It is easly to see that if is an *averaged* mapping, then is nonexpansive.

Proposition 4 (see [20]). *Let be a given mapping. Then consider the following.*(i) is nonexpansive if and only if the complement is -*ism*. (ii) is averaged if and only if the complement is *-ism* for some . Indeed, for is *-averaged* if and only if is *-ism*. (iii)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 , where , then the composite is *-averaged*, where .

In this paper, we use an equilibrium bifunction for solving the equilibrium problems, let us assume that satisfies the following conditions: (A1); (A2) is monotone, that is, ;(A3)for each is weakly upper semicontinuous; (A4)for each is convex; semicontinuous.

Lemma 5 (see [6]). *Assume that satisfies (A1)–(A4). For and , define a mapping as follows:
**
for all . Then, the following hold: *(i)* is single-valued; *(ii)* is firmly nonexpansive, that is, for any ;*(iii)*;
*(iv)* is closed and convex. *

Lemma 6 (see [32]). *Let be a closed convex subset of a real Hilbert space . Let and be two mappings from satisfying (A1)–(A4) and let and are defined as in Lemma 5 associated to and , respectively. For given is a solution of problem (2) if and only if is a fixed point of the mapping defined by
**
where . *

Lemma 7 (see [33]). *Let and be sequences of nonnegative real numbers satisfying the inequality
**
If and , then exists. If, in addition, has a subsequence which converges to zero, then .*

#### 3. Weak Convergence Theorem

In this section, we prove a weak convergence theorem by an extragradient methods for finding a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems in Hilbert spaces. The function is a continuous differentiable function with the minimization problem given by In 2010, Xu [17] considered the following Tikhonov regularized problem: where is the regularization parameter. The gradient given by is -Lipschitz continuous and -strongly monotone (see [29] for the details).

Lemma 8 (see [17, 29]). *The following hold: *(i)* for any , where and denote the set of fixed points of and the solution set of VIP; *(ii)* is -averaged for each , where . *

Theorem 9. *Let be a nonempty closed convex subset in a real Hilbert space . Let , and be two bifunctions from satisfying –. Let be a quasi-nonexpansive mapping of into itself such that be demiclosed at zero, that is, if and , then , with . Let , , , and be the sequence in generated by the following extragradient algorithm:
**
where , for some and for some . Then, the sequences and converge weakly to an element . *

* Proof. *By Lemma 8, we have that is -averaged for each , where . Hence, by Proposition , is nonexpansive. From and , we have . Without loss of generality, we assume that . Hence, for each is -averaged with
This implies that is nonexpansive for all .

Next, we show that the sequence is bounded. Indeed, take a fixed arbitrarily. Let and be defined as in Lemma 5 associated to and , respectively. Thus, we get and . Put and . From (29), we have
This implies that . Thus, we obtain . Put for each . Then, by Proposition 1(ii), we have
Hence, by Proposition 1(i), we have
So, we have
Then, from the last inequality we conclude that
where and . Since and for some , we conclude that and . Therefore, by Lemma 7, we note that exists for each and hence the sequences , and are bounded. From (22), we also obtain
where and . Since exists and , it follows that
Similarly, from inequality (22), we have
where and . Since exists and , we obtain
Moreover, we note that
From (26) and , it is implied that
Note that . This together with (24) and (28) implies that . Also, from , it follows that . Since is a Lipschitz condition, where is the adjoint of , we have . Since is a bounded sequence, there exists a subsequence of that converges weakly to some .

Next, we show that . Since and , it is known that and . Let be a set value mappings defined by
where . Hence, by [34], is maximal monotone and if and only if . Let . Then we have and hence . So, we obtain . On the other hand, from and , we have
and hence
Therefore, from and , we get
By taking , we obtain . Since is maximal monotone, it follows that and hence . Therefore, by Lemma 8, .

Next, we show that . Since and , it follows by the demiclosed principle that . Hence, we have .

Next, we show that . Let be a mapping which is defined as in Lemma 6, thus we have
and hence
By taking , we have . From and , we obtain . According to demiclosedness and Lemma 6, we have . Therefore, we have . Let be another subsequence of such that . We show that , suppose that . Since exists for all , it follows by the Opial's condition that
It is a contradiction. Thus, we have and so . Further, from , it follows that and hence , . This completes the proof.

Theorem 9 extends the extragradient method according to Nadezhkina and Takahashi [35].

Corollary 10. *Let be a nonempty closed convex subset in a real Hilbert space . Let and be two bifunctions from satisfying (A1)–(A4). Let and let and be defined as in Lemma 5 associated to and , respectively. Let be a quasi-nonexpansive mapping of into itself such that . Let , , , and be the sequence in generated by the following extragradient algorithm:
**
where , for some and for some . Then, the sequences and converge weakly to an element . *

Corollary 11. *Let be a nonempty closed convex subset in a real Hilbert space . Let and be two bifunctions from satisfying (A1)–(A4). Let and let and be defined as in Lemma 5 associated to and , respectively. Let be a quasi-nonexpansive mapping of into itself such that . Let , , and be the sequence in generated by the following extragradient algorithm:
**
where , for some and for some . Then, the sequences and converge weakly to an element . *

* Proof. *Setting in Theorem 9, we obtain the desired result.

Corollary 12. *Let be a nonempty closed convex subset in a real Hilbert space . Let be a quasi-nonexpansive mapping of into itself such that . Let , , and be the sequence in generated by the following extragradient algorithm:
**
where , for some and for some . Then, the sequences and converge weakly to an element . *

* Proof. *Setting in Theorem 9, we obtain the desired result.

#### Acknowledgments

The authors would like to thank The Commission on Higher Education for financial support. Moreover, K. Sombut is also supported by The Strategic Scholarships Fellowships Frontier Research Networks under Grant CHE-Ph.D-THA-SUP/86/2550, Thailand.