Abstract

The purpose of this paper is to introduce the extragradient methods for solving split feasibility problems, generalized equilibrium problems, and fixed point problems involved in nonexpansive mappings and pseudocontractive mappings. We establish the results of weak and strong convergence under appropriate conditions. As applications of our three main theorems, when the mappings and their domains take different types of cases, we can obtain nine iterative approximation theorems and corollas on fixed points, variational inequality solutions, and equilibrium points.

1. Introduction

Let and be two real Hilbert spaces, and let and be two nonempty closed and convex subsets of and , respectively. Let be a bounded linear operator with its adjoint . The split feasibility problem is to find a point such that

We denote the solution set of the split feasibility problem by

Problem (1) was first introduced by Censor and Elfving [1] in the finite-dimensional spaces and further has been studied by many researchers (see, for example, [2–6]) and the references therein. To solve the , Byrne [2, 7] first introduced the so-called CQ algorithm as follows:where , denotes the projection onto , and is the spectral radius of the self-adjoint operator . Many authors continue to study the CQ algorithm in its various forms (see, for example, [8–14]). The CQ algorithm can be viewed from two different but equivalent ways: optimization and fixed point [6]. From the view of optimization point, in (2) if and only if is a solution of the following minimization problem with zero optimal value , where is a differentiable convex function and has a Lipschitz gradient given by , with Lipschitz constant . Thus, solves the if and only if solves the variational inequality problem of finding such that for all .

Xu [6] considered the following Tikhonov regularized problem:where is the regularization parameter. We observe that the gradientis -Lipschitz continuous and -strongly monotone. The fixed point approach method to solve the is based on the following observations. Let , and assume that . Then, , which implies that , and thus, . Hence, we have the fixed point equation . Requiring that , we consider the fixed point equation

In [6], it is proved that the solutions of fixed point equation (6) are precisely the solutions of the .

Let be a nonlinear mapping and be a bifunction from to , where is the set of real numbers. The generalized equilibrium problem is to find such that . The set of solutions is denoted by . If , then is denoted by . If for all , then is denoted by . This is the set of solutions of the variational inequality for (see, for example, [15–21]). If , then where .

In 2008, Takahashi and Takahashi [15] have suggested the following iterative method. Let be a sequence generated by

Under some appropriate conditions, they proved that the sequence converges strongly to a point .

Motivated and inspired by the above works, we will investigate the weak and strong convergence methods for solving the split feasibility problems, generalized equilibrium problems, and fixed point problems involved in nonexpansive mappings and pseudocontractive mappings. As applications of our three main theorems, when the mappings and their domains take different types of cases, we can obtain nine iterative approximation theorems and corollaries on fixed points, variational inequality solutions, and equilibrium points. So, our results in this paper generalize and improve upon the corresponding modern results of many other authors.

2. Preliminaries

Let be a real Hilbert space with the inner product and norm and be a nonempty, closed, and convex subset of . Recall that a mapping is said to be monotone if for all [18, 19]. A mapping is said to be -strongly monotone whenever there exists a positive real number such that for all . A mapping is said to be -inverse strongly monotone if there exists a positive real number such that for all . Recall that the classical variational inequality problem, which we denote by , is to find such that , for all [16, 17]. It is well known that, for any , there exists a unique nearest point in , denoted by , such that . It is well known that is a nonexpansive and monotone mapping from onto and satisfy the following:(1) for all (2) for all (3)The relation holds for all

Let be a monotone mapping of into . In the context of the variational inequality problem, it is easy to see from (2) that

For solving the equilibrium problem, we assume that satisfies the following conditions:(i) for all (ii) is monotone, that is, for all (iii) for each , (iv) for each , the function is convex and lower semicontinuous

If for every , we see that the equilibrium problem is reduced to the variational inequality problem.

Lemma 1 (see [22]). Let be a nonempty, closed, and convex subset of , and let be a bifunction from to satisfying . For and , consider the mapping defined by

Then, for all , is single-valued, is closed and convex, , and is firmly nonexpansive, that is, for all .

Lemma 2 (see [23]). Let be a nonempty, closed, and convex subset of , be a bifunction from to satisfying , ad be a multivalue mapping from into itself defined by whenever and otherwise. Then, is a maximal monotone operator with the domain , for all and .

Definition 1. Let be a nonlinear operator.(1) is said to be -Lipschitz whenever there exists such that . If , we call is nonexpansive, and is said to be a contraction if .(2) is said to be firmly nonexpansive if is nonexpansive and is the identity mapping, or equivalently, . Alternatively, is firmly nonexpansive if and only if can be expressed as , where is nonexpansive.(3) is said to be -averaged nonexpansive mapping, if there exists a nonexpansive mapping , such that , where . Thus, firmly nonexpansive mappings are -averaged mapping.(4) is said to be pseudocontractive if and only if .(5) is said to be -strictly pseudocontractive if and only if there exists , such that

Remark 1 (see [2]). Let be a given mapping:(i) is nonexpansive if and only if the complement is -inverse strongly monotone.(ii)If is -inverse strongly monotone, then for is -inverse strongly monotone.(iii) is averaged if and only if the complement is -inverse strongly monotone for some . Indeed, for is -averaged if and only if is -inverse strongly monotone.We denote by the set of fixed points of . Note that every -inverse strongly monotone mapping is Lipschitz and . Every nonexpansive mapping is a -strictly pseudocontractive mapping and every -strictly pseudocontractive mapping is pseudocontractive. Assume that is a strictly pseudocontractive. If , we easily find that is -inverse strongly monotone and . Note that is pseudocontractive if and only if is monotone, and . There are a lot works associated with the fixed point algorithms for nonexpansive mappings and pseudocontractive mappings (see, for example, [24–28]).
A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if the graph of is not properly contained in the graph of any other monotone mappings. Also, a monotone mapping is maximal if and only if, for for every implies . Let be an inverse strongly monotone mapping and let be the normal cone to at , i.e., . DefineIt is known that is maximal monotone and if and only if [29, 30].

Lemma 3 (see [8]). Let and be nonempty, closed, and convex subsets of real Hilbert spaces and , respectively, and let be a bounded linear operator and be a continuous differentiable function. If and , then(1) is -inverse strongly monotone mapping(2) is -averaged(3) is -averaged, with (4) is nonexpansive

Lemma 4 (see [31]). Let be a real Hilbert space, be a closed convex subset of , and be a continuous pseudocontractive mapping. Then,(i) is a closed convex subset of (ii) is demiclosed at zero, i.e., if is a sequence in such that and ; as , then .

Lemma 5 (see [32]). Let be a real Hilbert space. Then, for all and , for such that , the following equality holds:

Lemma 6 (see [33]). Let be a nonempty closed and convex subset of a real Hilbert space and be a nonexpansive mapping. Then, is demiclosed at zero.

Lemma 7 (see [34]). Let and be sequences of nonnegative real numbers satisfying . If converges, then exists.

Lemma 8 (see [35]). Let be a nonempty closed convex subset of a real Hilbert space and let be a -strictly pseudocontraction with a fixed point. Define by for each . Then, as is nonexpansive such that .

Lemma 9 (see [36]). Let be a sequence of nonnegative real numbers satisfying , where and is a sequence such that , or , and where . Then, .

Lemma 10 (see [37]). Let , , and be the sequences in such that, and there exists a real number with for all . Then, the following holds:(i), where (ii)There exists such that

Lemma 11 (see [31]). Let be a real Hilbert space. Then, for any given , the following inequality holds: .

3. Weak and Strong Convergence Results

Now, we are ready to state and prove some of our main results in this section.

Theorem 1. Assume that and are 2 nonempty, closed, and convex subsets of real Hilbert spaces and , respectively. Let be a bounded linear operator, be a continuous differentiable function, be a bifunction from to satisfying , be an -inverse strongly monotone mapping from into , be a nonexpansive mapping, and be a strictly pseudocontractive mapping with constant such that . Let , , , and be sequences generated by the following extragradient algorithm:where and . Suppose the following conditions are satisfied:(a), , (b) for some (c) for some (d) and ,(e)Then, converges strongly to the point provided .

Proof. For any fixed , we find that for and . We see from Lemma 8 that is nonexpansive and . It is observed that can be rewritten as . From condition (e) and Lemma 1, we haveFrom (14), (15), and Lemma 3, it follows thatBy the property of metric projection, we haveFurthermore, by the property of metric projection, we haveHence, we haveSo, from (15), we obtainWe find from (14) and (16) and the last inequality thatConsequently, from condition (a), we deduce that is bounded and so there exist the sequences , , and . Put for all . We find from (15), (16), (19), and Lemma 5 thatFrom (14) and the last inequality, we conclude thatThis yields thatSince , we haveFrom (97) and the condition (a)–(d), we also obtainIt is observe thatUsing Lemma 1 and (14), we haveIt follows thatFrom (19) and (29), we find thatFrom (14) and the last inequality, we conclude thatThis yields thatIt follows from condition (a) and thatSince , , , , we obtain as . Note that . This implies thatAlso, from , , , and , we getSince is Lipschitz continuous, we obtain .
Next, we show thatwhere . To show it, choose a subsequence of such thatSince is bounded, there exists a subsequence of , converges weakly to . Without loss of generality, we assume that . Since , as , we obtain that . Since and is closed and convex, we obtain . First, we show that . Then, from (34), (35), Lemma 6, and Lemma 4, we have that . We now show that . By , we know thatIt follows from thatHence,For with and , let . Since and , we obtain . So, from (74), we haveSince , we have . Furthermore, from the inverse strongly monotonicity of , we have . It follows from condition and and , we haveas . From and , we haveand hence,Letting , we have, for each ,This implies that . Next, we show that (1). LetThen, is maximal monotone and if and only if [29]. Let be the graph of , let . Then, we have and hence . Therefore, we have for all . By the property of metric projection, from and , we have , and hence,From for all and , we haveThus, we obtain as . Since is maximal monotone, we have , and hence, . This implies . This implies that . Thanks to (37), we arrive atNext, we show that as . Observe thatWith the help of (14), we obtainwhich implies thatIt follows from condition (a) and Lemma 9 thatTherefore, from , , we can conclude that , , , and converge strongly to the same point . The proof is complete.