Journal of Applied Mathematics

Volume 2014 (2014), Article ID 280158, 8 pages

http://dx.doi.org/10.1155/2014/280158

Research Article

Convergence Theorems for Hierarchical Fixed Point Problems and Variational Inequalities

Ibrahim Karahan¹ and Murat Ozdemir²

¹Department of Mathematics, Faculty of Science, Erzurum Technical University, 25240 Erzurum, Turkey

²Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erzurum, Turkey

Received 17 March 2014; Accepted 16 June 2014; Published 6 July 2014

Academic Editor: Luigi Muglia

Copyright © 2014 Ibrahim Karahan and Murat Ozdemir. 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.

This paper deals with a modi…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 ,  .

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 [1–4]). 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 [5–12]). 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 ,  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 of  .

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, Opial’s 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 or  .

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.

Table 1: The numerical results are obtained by using FORTRAN90 Programming Language.

Conflict of Interests

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

References

D. R. Sahu, “Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces,” Commentationes Mathematicae Universitatis Carolinae, vol. 46, no. 4, pp. 653–666, 2005.

R. P. Agarwal, D. O'Regan, and D. R. Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive mappings,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 61–79, 2007.

D. R. Sahu, S. M. Kang, and V. Sagar, “Iterative methods for hierarchical common fixed point problems and variational inequalities,” Fixed Point Theory and Applications, vol. 2013, article 299, 2013.

D. R. Sahu, S. M. Kang, and V. Sagar, “Approximation of common fixed points of a sequence of nearly nonexpansive mappings and solutions of variational inequality problems,” Journal of Applied Mathematics, vol. 2012, Article ID 902437, 12 pages, 2012.

Y. Yao, Y. J. Cho, and Y. Liou, “Iterative algorithms for hierarchical fixed points problems and variational inequalities,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1697–1705, 2010.

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

A. Moudafi, “Krasnoselski-Mann iteration for hierarchical fixed-point problems,” Inverse Problems, vol. 23, no. 4, pp. 1635–1640, 2007.

F. Cianciaruso, V. Colao, L. Muglia, and H. K. Xu, “On an implicit hierarchical fixed point approach to variational inequalities,” Bulletin of the Australian Mathematical Society, vol. 80, no. 1, pp. 117–124, 2009.

G. Marino and H.-K. Xu, “Explicit hierarchical fixed point approach to variational inequalities,” Journal of Optimization Theory and Applications, vol. 149, no. 1, pp. 61–78, 2011.

H. K. Xu, “Viscosity method for hierarchical fixed point approach to variational inequalities,” Taiwanese Journal of Mathematics, vol. 14, no. 2, pp. 463–478, 2010.

W. Q. Deng, “New viscosity method for hierarchical fixed point approach to variational inequalities,” Fixed Point Theory and Applications, vol. 2013, article 219, 2013.

L. C. Ceng and A. Petrusel, “Krasnoselski-Mann iterations for hierarchical fixed point problems for a finite family of nonself mappings in Banach spaces,” Journal of Optimization Theory and Applications, vol. 146, no. 3, pp. 617–639, 2010.

G. Marino and H. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.

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.

Y. Wang and W. Xu, “Strong convergence of a modified iterative algorithm for hierarchical fixed point problems and variational inequalities,” Fixed Point Theory and Applications, vol. 2013, article 121, 9 pages, 2013.

N. C. Wong, D. R. Sahu, and J. C. Yao, “A generalized hybrid steepest-descent method for variational inequalities in Banach spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 754702, 28 pages, 2011.

A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.

H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.

M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,” Nonlinear Analysis, vol. 73, no. 3, pp. 689–694, 2010.

T. Suzuki, “Moudafi's viscosity approximations with Meir-Keeler contractions,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 342–352, 2007.

J. G{\'o}rnicki, “Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces,” Commentationes Mathematicae Universitatis Carolinae, vol. 30, no. 2, pp. 249–252, 1989.

I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 of Studies in Computational Mathematics, pp. 473–504, North-Holland Publisher, Amsterdam, The Netherlands, 2000.