#### Abstract

This paper deals with a modi*ā¦*fied iterative projection method for approximating a solution of hierarchical fixed point problems for nearly nonexpansive mappings. Some strong convergence theorems for the proposed method are presented under certain approximate assumptions of mappings and parameters. As a special case, this projection method solves some quadratic minimization problem. It should be noted that the proposed method can be regarded as a generalized version of Wang and Xu (2013), Ceng et al. (2011), Sahu et al. (2012), and many other authors.

#### 1. Introduction

Throughout this paper, is a nonempty closed convex subset of a real Hilbert space ; denotes the associated inner product; stands for the corresponding norm. Let be the identity mapping on and the metric projection of onto . To begin with, let us recall the following concepts which are commonly used in the context of convex and nonlinear analysis. For all , , a mapping is said to be monotone if , -strongly monotone if there exists a positive real number such that , and -Lipschitzian if there exists a positive real number such that . Let us also recall that a mapping is said to be contraction if there exists a constant such that , nonexpansive if , and asymptotically nonexpansive if for each there exists a positive constant with such that , for all ,ā .

As a generalization of asymptotically nonexpansive mappings, Sahu [1] introduced the class of nearly Lipschitzian mappings. Let us fix a sequence in with . A mapping is called nearly Lipschitzian with respect to if for each there exists a constant such that The infimum constant for which (1) holds will be denoted by and called nearly Lipschitz constant. Notice that A nearly Lipschitzian mapping with sequence is said to be nearly nonexpansive if for . In this paper, we study a nearly nonexpansive mapping which is studied by some authors (see [1ā4]). This type of mappings (not necessarily continuous) is important with regard to generalization of the asymptotically nonexpansive mappings. Particularly, the class of this type of mappings is an intermediate class between the class of asymptotically nonexpansive mappings and that of mappings of asymptotically nonexpansive type (please see [1] for nearly nonexpansive mappings examples).

Now, we focus on the hierarchical fixed point problem for a nearly nonexpansive mapping with respect to a nonexpansive mapping . This problem is to find a point satisfying where is the set of fixed points of ; that is, . It is easy to see that the problem (3) is equivalent to the problem of finding a point that satisfies .

Let be the normal cone to defined by Then, the hierarchical fixed point problem is equivalent to the variational inclusion problem which is to find a point such that The existence problem of hierarchical fixed points for a nonlinear mapping and approximation problem has been studied by several authors (see [5ā12]). In 2006, Marino and Xu [13] introduced the following viscosity iterative method: where is a contraction, is a nonexpansive mapping, and is a strongly positive bounded linear operator on ; that is, , for some . Under the appropriate conditions, they proved that the sequence defined by (6) converges strongly to the unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for ; that is, for all .

In 2011, Ceng et al. [14] generalized the iterative method (6) of Marino and Xu [13] by taking a Lipschitzian mapping and Lipschitzian and strongly monotone operator instead of the mappings and , respectively. They gave the following iterative method: where is a metric projection and is a nonexpansive mapping, and they also proved that the sequence generated by (9) converges strongly to the unique solution of the variational inequality Recently, motivated by the iteration method (9) of Ceng et al. [14], Wang and Xu [15] studied the following iterative method for a hierarchical fixed point problem: where , are nonexpansive mappings, is a -Lipschitzian mapping, and is a -Lipschitzian and -strongly monotone operator. They proved that under some suitable assumptions on the sequences and , the sequence generated by (11) converges strongly to the hierarchical fixed point of with respect to the mapping which is the unique solution of the variational inequality (10). With this study, Wang and Xu extend and improve the many recent results of other authors.

Let be a sequence of mappings from into and fix a sequence in with . Then, is called a sequence of nearly nonexpansive mappings [16] with respect to a sequence if It is obvious that the sequence of nearly nonexpansive mappings is a wider class of sequence of nonexpansive mappings. For the sequence of nearly nonexpansive mappings defined by (12), Sahu et al. [3] introduced a new iteration method to solve the hierarchical fixed point problem and variational inequality problem.

*Remark 1. *Let be a sequence of nearly nonexpansive mappings. Then,(1)for each ,ā is not a nearly nonexpansive mapping,(2)if is a mapping on defined by for all , then it is clear that is a nonexpansive mapping.

Recently, in 2012, Sahu et al. [4] introduced the following iterative method for the sequence of nearly nonexpansive mappings defined by (12): They proved that the sequence generated by (13) converges strongly to the unique solution of the variational inequality (10).

*Remark 2. *Since the mapping is not a nearly nonexpansive mapping for each , if one takes for all such that is a nearly nonexpansive mapping, then the iteration (13) is not well defined. Hence, the main result of Sahu et al. [4] is no longer valid for a nearly nonexpansive mapping.

In this paper, motivated and inspired by the work of Wang and Xu [15], we introduce a modified iterative projection method to find a hierarchical fixed point of a nearly nonexpansive mapping with respect to a nonexpansive mapping. We show that our iterative method converges strongly to the unique solution of the variational inequality (10). As a special case, the presented projection method solves some quadratic minimization problem. Also, our method improves and generalizes corresponding results of Yao et al. [5], Marino and Xu [13], Ceng et al. [14], Wang and Xu [15], Moudafi [17], Xu [18], Tian [19], and Suzuki [20].

#### 2. Preliminaries

This section contains some lemmas and definitions which will be used in the proof of our main result in the following section. We write to indicate that the sequence converges weakly to and for the strong convergence. A mapping is called a metric projection if there exists a unique nearest point in denoted by such that It is easy to see that is a nonexpansive mapping and it satisfies the following inequality: Now, we give the definitions of a demicontinuous mapping, asymptotic radius, and asymptotic center.

Let be a nonempty subset of a Banach space and a mapping. is called demicontinuous if in implies in .

Let be a nonempty closed convex subset of a uniformly convex Banach space , a bounded sequence in , and a functional defined by The infimum of over is called asymptotic radius of with respect to and is denoted by . A point is said to be an asymptotic center of the sequence with respect to if The set of all asymptotic centers is denoted by . Related with these definitions, we will use the following in our main results.

Theorem 3 (see [21]). *Let be a nonempty closed convex subset of a uniformly convex Banach space satisfying the Opial condition. If is a sequence in such that , then is the asymptotic center of in .*

Lemma 4 (see [1]). *Let be a nonempty closed convex subset of a uniformly convex Banach space and a demicontinuous nearly Lipschitzian mapping with sequence such that . If is a bounded sequence in such that
**
then is a fixed point ofā .*

Lemma 5 (see [14]). *Let be a -Lipschitzian mapping and let be a -Lipschitzian and -strongly monotone operator; then, for ,
**
That is to say, the operator is -strongly monotone.*

Lemma 6 (see [22]). *Let be a nonempty subset of a real Hilbert space . Suppose that and . Let be a -Lipschitzian and -strongly monotone operator. Define the mapping by
**
Then, is a contraction that provided . More precisely, for ,
**
where *

Lemma 7 (see [10]). *Assume that is a sequence of nonnegative real numbers such that
**
where and are sequences of real numbers which satisfy the following conditions:*(i)*, and ;*(ii)*either or .**Then .*

#### 3. Main Result

Theorem 8. *Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping and a demicontinuous nearly nonexpansive mapping on with respect to the sequence such that . Let be a -Lipschitzian mapping and a -Lipschitzian and -strongly monotone operator such that the coefficients satisfy , , where . For an arbitrarily initial value , consider the sequence in generated by
**
where and are sequences in satisfying the following conditions:*(i)*, and ;*(ii)*, , , and ;*(iii)*, and , .**Then, the sequence converges strongly to , where is the unique solution of the variational inequality (10).*

*Proof. *Since the mapping is a strongly monotone operator from Lemma 5, it is known that the variational inequality (10) has a unique solution. Let us denote this solution by . Now, we divide our proof into five steps.*Step **1*. First we show that the sequence generated by (23) is bounded. From condition (ii), without loss of generality, we may suppose that , for all . Hence, we get . Let and . Then we have
and, by using the definition of nearly nonexpansive mapping and Lemma 6, we obtain
From (24) and (25), we get
From condition (ii), this sequence is bounded, and so we can write
where
Therefore, by induction, we get
Hence, we obtain that is bounded. So, the sequences , , , , and are bounded too.*Step **2*. Secondly, we show that . By using the iteration (23), we have
where is a constant such that , and
So, from (30) and (31), we get
where
By using the conditions (ii) and (iii), since , it follows from Lemma 7 that
*Step **3*. Next, we show that . For , we get
and so
Hence, we obtain from (35) and (36) that
Since and are bounded, it follows from (34), (37), condition (i), and condition (iii) that
Combining (38) and condition (iii), we have
*Step **4*. Now, we show that , where is the unique solution of the variational inequality (10). Since the sequence is bounded, it has a weak convergent subsequence such that
Let , as . Then, Opialās condition guarantees that the weakly subsequential limit of is unique. Hence, this implies that , as . So, it follows from (38), Theorem 3, and Lemma 4 that . Therefore
*Step **5*. Finally, we show that the sequence converges strongly to . By using inequality (15), we have
Also, by using inequality (24), we get
which implies that
where
From condition (ii), by using Step 4, we get
So, it follows from Lemma 7 that the sequence generated by (23) converges strongly to which is the unique solution of the variational inequality (10). This completes the proof.

*Remark 9. *In particular, the point is the minimum norm fixed point of ; namely, is the unique solution of the quadratic minimization problem
Indeed, since the point is the unique solution of the variational inequality (10), if we take and , then we get
So we have
Hence, is the unique solution to the quadratic minimization problem (47).

Since a nearly nonexpansive mapping can be reduced to a nonexpansive mapping by taking the sequence as a zero sequence, under the appropriate changes on the control sequences arising from Lemma 7, we can derive main results of Wang and Xu [15, Theorem 3.1] and Ceng et al. [14, Theorem 3.1] as the following corollaries.

Corollary 10. *Let be a nonempty closed convex subset of a real Hilbert space . Let , be nonexpansive mappings such that . Let be a -Lipschitzian mapping and a -Lipschitzian and -strongly monotone operator such that these coefficients satisfy , , where . For an arbitrarily initial value , consider the sequence in generated by (11) where and are sequences in satisfying the following conditions:*(i)*, and ;*(ii)*;*(iii)*, and .**Then, the sequence converges strongly to , where is the unique solution of the variational inequality (10). In particular, the point is the minimum norm fixed point of ; that is, is the unique solution of the quadratic minimization problem (47).*

*Proof. *In the proof of Theorem 8, for all , if we take the mapping as a nonexpansive mapping, then the desired conclusion is obtained.

Corollary 11. *Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping such that . Let be a -Lipschitzian mapping with being a -Lipschitzian and -strongly monotone operator such that these coefficients satisfy , , where For an arbitrarily initial value , consider the sequence in generated by (9) where and are sequences in satisfying the following conditions:*(i)*, and ;*(ii)*either orāā.**Then, the sequence converges strongly to , where is the unique solution of the variational inequality (10).*

*Proof. *In the proof of Theorem 8, let where is the identity mapping and a nonexpansive mapping. Then, the proof is clear.

Let be a sequence of mappings from into and let be a sequence in with , for each . Then, is called a sequence of nearly nonexpansive mappings with respect to a sequence if Considering Remarks 1 and 2, it is too easy to see that the sequence defined by (50) is different from the sequence defined by (12) of Wong et al. [16]. The iterative scheme defined by (23) can be modified for the sequence of nearly nonexpansive mappings defined by (50). Accordingly, the problem written in the following remark arises.

*Remark 12. *For a sequence of nearly nonexpansive mappings (50) with a nonempty common fixed points set, it is an open problem whether or not an iteration process generated by this sequence will converge strongly to a common fixed point.

*Example 13. *Let and . Let , , , and
for all . It is clear that is nonexpansive, is -Lipschitzian with , is -Lipschitzian and -strongly monotone operator with , and is a nearly nonexpansive mapping with respect to the sequence . Define sequences and in by and , and take , , and . It is easy to see that all the conditions of Theorem 8 are satisfied. So, the sequence generated by the iterative scheme (23) becomes
and it converges strongly to which is the unique fixed point of and the unique solution of the variational inequality (10) over . The first ten values of iterative scheme (52) for the different initial values , , and are as in Table 1.

#### Conflict of Interests

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