Research Article | Open Access

# Algorithms for Solving the Variational Inequality Problem over the Triple Hierarchical Problem

**Academic Editor:**D. Anderson

#### Abstract

This paper discusses the monotone variational inequality over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence theorem for the proposed algorithm to the solution is guaranteed under some suitable assumptions.

#### 1. Introduction

Let be a closed convex subset of a real Hilbert space with the inner product and the norm . We denote weak convergence and strong convergence by notations and , respectively.

A mapping is said to be *monotone* if is said to be -*strongly monotone* if there exists such that . is said to be -*inverse-strongly monotone* if there exists such that . is said to be -*Lipschitz continuous* if there exists such that . A linear bounded operator is said to be *strongly positive* on if there exists with the property .

Let be a -*contraction* if there exists such that

Let be *nonexpansive* such that
A point is a *fixed point* of provided . Denote by the set of fixed points of ; that is, . If is bounded closed convex and is a nonexpansive mapping of into itself, then is nonempty (see [1]). Let be a nonlinear mapping. The *Hartmann-Stampacchia variational inequality* [2] is to finding such that
The set of solutions of (1.3) is denoted by . The variational inequality has been extensively studied in the literature [3, 4].

We discuss the following variational inequality problem over the fixed point set of a nonexpansive mapping (see [5–12]), which is called the *hierarchical problem*. Let a monotone, continuous mapping and a nonexpansive mapping .
This solution set is denoted by .

We introduce the following variational inequality problem over solution set of variational inequality problem and the fixed point set of a nonexpansive mapping (see [13–16]), which is called *the triple hierarchical problem* (or *the triple hierarchical constrained optimization problem* (see also [13])). Let an inverse-strongly monotone , a strongly monotone and Lipschitz continuous , and a nonexpansive mapping .
where .

In 2009, Iiduka [13] introduced an iterative algorithm for the following *triple hierarchical constrained optimization problem*, the sequence defined by the iterative method below, with the initial guess is chosen arbitrarily,
where and satisfies certain conditions. Let be an inverse-strongly monotone, be a strongly monotone and Lipschitz continuous, and be a nonexpansive mapping, then the sequence converges to strong analysis on (1.6).

In 2011, Ceng et al. [17] studied the new following algorithms. For is chosen arbitrarily, they defined a sequence iterative by where the mapping are nonexpansive mappings with . Let be a Lipschitzian and strongly monotone operator and be a contraction mapping satisfied some conditions. They proved that the proposed algorithms strongly converge to the minimum norm fixed point of .

Very recently, Yao et al. [18] studied the following algorithms. For is chosen arbitrarily, let the sequence be generated iteratively by where the sequences and are two sequences in . Then converges strongly to the unique solution of the variational inequality as follows. Find a point such that where is a strongly positive linear bounded operator, is a -contraction, and is a nonexpansive mapping satisfied some suitable conditions. The solution (1.9) is denoted by .

In this paper, we introduce a new iterative algorithm for solving the triple hierarchical problem, which contain algorithms (1.6) and (1.8) as follows: The strong convergence for the proposed algorithms to the solution is solved under some assumptions. Our results generalize and improve the results of Ceng et al. [17], Iiduka [13], Yao et al. [18], and some authors.

#### 2. Preliminaries

Let be a real Hilbert space and be a nonempty closed convex subset of . The metric (or nearest point) projection from onto is the mapping which assigns to each point the unique point in satisfying the property The following properties of projection are useful and pertinent to our purposes.

Lemma 2.1. *Given and ,*(a)*,
*(b)*,
*(c)* is a firmly nonexpansive mapping of onto and satisfies
**Consequently, is nonexpansive and monotone.*

Lemma 2.2. *There holds the following inequality in an inner product space *

Lemma 2.3 (see [19]). *Let be a closed convex subset of a real Hilbert space and let be a nonexpansive mapping. Then is demiclosed at zero, that is,
**
implies .*

Lemma 2.4 (see [20]). *Each Hilbert space satisfies Opial’s condition, that is, for any sequence with , the inequality
**
hold for each with .*

Lemma 2.5 (see [21]). *Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .*

Lemma 2.6 (see [10]). *Let be -strongly monotone and -Lipschitz continuous and let . For , define by for all . Then, for all ,
**
hold, where .*

Lemma 2.7 (see [22]). *Assume is a sequence of nonnegative real numbers such that
**
where and is a sequence in such that*(i)*,*(ii)* or .**Then .*

*Remark 2.8. *If is a strongly positive linear bounded operator and is a -contraction, then for , the mapping is strongly monotone. In fact, we have

#### 3. Main Results

In this section, we introduce a new iterative algorithm for solving monotone variational inequality problem (where is a strongly positive linear bounded operator, is a -contraction) over solution set of variational inequality problem over the fixed point set of a nonexpansive mapping.

Theorem 3.1. *Let be a nonempty closed and convex subset of a real Hilbert space . Let be a strongly positive linear bounded operator, be a -contraction, and be a positive real number such that . Let be -Lipschitzian and -strongly monotone operators with constant and , respectively. Let be a nonexpansive mapping with . Let and , where . Assume that , where . Suppose is a sequence generated by the following algorithm arbitrarily and
**
where satisfy the following conditions:** and ;**;**;**, and .**Then the sequence converges strongly to , which is the unique solution of another variational inequality
*

*Proof. * We will divide the proof into four steps.*Step 1.* We will show is bounded. For any , we have
From (3.1), we deduce that
Substituting (3.3) into (3.4), we obtain
By induction, it follows that
Therefore, is bounded and so are , and .*Step 2.* We will show that , and . From (3.1), we have
It follows that
where is a constant such that

By the conditions ()–() allow us to apply Lemma 2.7, we get
On the other hand, we note that
by (), it follows that
From (3.7), we observe that
It follows that
From the conditions ()–() and the boundedness of , and , which implies that
Hence, by Lemma 2.5, we have
From (3.12) and (3.16), we obtain
*Step 3.* We will show that is proven. Choose a subsequence of such that
The boundedness of implies the existences of a subsequence of and a point such that converges weakly to . We may assume without loss of generality that . Assume . Since with guarantee that
which has a contradiction. Therefore, . Since , then , it follows that
Setting and by (), we notice that
Hence, we get
Next we will show that is proven. Choose a subsequence of such that
The boundedness of implies the existences of a subsequence of and a point such that converges weakly to . We may assume without loss of generality that . Assume . By with guarantee that
which has a contradiction. Therefore, . From , we compute
Using (3.10), we get
*Step 4.* Finally, we prove . We observe that
From (3.1), we compute
Since , and are all bounded, we can choose a constant such that
It follows that
where
By the conditions (), (), (3.22), and (3.26), we get
Now, applying Lemma 2.7 and (3.30), we conclude that . This completes the proof.

Next, the following example shows that all conditions of Theorem 3.1 are satisfied.

*Example 3.2. *For instance, let and . Then, clearly the sequences , satisfy the following condition ():
We will show that the condition () is achieved. Indeed, we have
The sequence satisfies the condition () by p-series. Next, we will show that the condition () is achieved. We compute
The sequence satisfies the condition (). Finally, we will show that the condition () is achieved. We compute
The sequence satisfies the condition ().

Corollary 3.3. *Let be a nonempty closed and convex subset of a real Hilbert space . Let be a strongly positive linear bounded operator, be a -contraction, and be a positive real number such that . Let be a nonexpansive mapping with . Assume that . Suppose is a sequence generated by the following algorithm arbitrarily and
**
where satisfy the following conditions ()–(). Then the sequence converges strongly to , which is the unique solution of variational inequality
*

*Proof. *Putting and in Theorem 3.1, we can obtain desired conclusion immediately.

Corollary 3.4. *Let be a nonempty closed and convex subset of a real Hilbert space . Let be a nonexpansive mapping with . Suppose is a sequence generated by the following algorithm arbitrarily and
**
where satisfy the following conditions ()–(). Then the sequence converges strongly to .*

*Proof. *Putting and in Corollary 3.3, we can obtain desired conclusion immediately.

*Remark 3.5. *Our results generalize and improve the recent results of Iiduka [13] and Yao et al. [18].

#### Acknowledgment

The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under Project NRU-CSEC no. 55000613) for financial support during the preparation of this paper.

#### References

- W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,”
*The American Mathematical Monthly*, vol. 72, pp. 1004–1006, 1965. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. Hartman and G. Stampacchia, “On some non-linear elliptic differential-functional equations,”
*Acta Mathematica*, vol. 115, pp. 271–310, 1966. View at: Publisher Site | Google Scholar | Zentralblatt MATH - F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “On a two-step algorithm for hierarchical fixed point problems and variational inequalities,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 208692, 13 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J.-C. Yao and O. Chadli, “Pseudomonotone complementarity problems and variational inequalities,” in
*Handbook of Generalized Convexity and Generalized Monotonicity*, vol. 76, pp. 501–558, Springer, New York, NY, USA, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. L. Combettes, “A block-iterative surrogate constraint splitting method for quadratic signal recovery,”
*IEEE Transactions on Signal Processing*, vol. 51, no. 7, pp. 1771–1782, 2003. View at: Publisher Site | Google Scholar - S. A. Hirstoaga, “Iterative selection methods for common fixed point problems,”
*Journal of Mathematical Analysis and Applications*, vol. 324, no. 2, pp. 1020–1035, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. Iiduka and I. Yamada, “A subgradient-type method for the equilibrium problem over the fixed point set and its applications,”
*Optimization*, vol. 58, no. 2, pp. 251–261, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - K. Slavakis and I. Yamada, “Robust wideband beamforming by the hybrid steepest descent method,”
*IEEE Transactions on Signal Processing*, vol. 55, no. 9, pp. 4511–4522, 2007. View at: Publisher Site | Google Scholar - K. Slavakis, I. Yamada, and K. Sakaniwa, “Computation of symmetric positive definite Toeplitz matrices by the hybrid steepest descent method,”
*Signal Processing*, vol. 83, no. 5, pp. 1135–1140, 2003. View at: Publisher Site | Google Scholar - 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*, vol. 8, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - I. Yamada, N. Ogura, and N. Shirakawa, “A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems,” in
*Inverse Problems, Image Analysis, and Medical Imaging*, vol. 313, pp. 269–305, American Mathematical Society, Providence, RI, USA, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - I. Yamada and N. Ogura, “Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings,”
*Numerical Functional Analysis and Optimization*, vol. 25, no. 7-8, pp. 619–655, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. Iiduka, “Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 12, pp. e1292–e1297, 2009. View at: Publisher Site | Google Scholar - H. Iiduka, “Iterative algorithm for solving triple-hierarchical constrained optimization problem,”
*Journal of Optimization Theory and Applications*, vol. 148, no. 3, pp. 580–592, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - T. Jitpeera and P. Kumam, “A new explicit triple hierarchical problem over the set of fixed points and generalized mixed equilibrium problems,”
*Journal of Inequalities and Applications*, vol. 2012, article 82, 2012. View at: Publisher Site | Google Scholar - N. Wairojjana and P. Kumam, “General iterative algorithms for hierarchical fixed points approach to variational inequalities,”
*Journal of Applied Mathematics*. Volume 2012, Article ID 174318, 20 pages. View at: Google Scholar - L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Iterative methods for triple hierarchical variational inequalities in Hilbert spaces,”
*Journal of Optimization Theory and Applications*, vol. 151, no. 3, pp. 489–512, 2011. View at: Publisher Site | Google Scholar - Y. Yao, Y.-C. Liou, and S. M. Kang, “Algorithms construction for variational inequalities,”
*Fixed Point Theory and Applications*, vol. 2011, Article ID 794203, 12 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,”
*Proceedings of Symposia in Pure Mathematics*, vol. 18, pp. 78–81, 1976. View at: Google Scholar - Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,”
*Bulletin of the American Mathematical Society*, vol. 73, pp. 591–597, 1967. View at: Publisher Site | Google Scholar | Zentralblatt MATH - T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,”
*Journal of Mathematical Analysis and Applications*, vol. 305, no. 1, pp. 227–239, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H.-K. Xu, “Iterative algorithms for nonlinear operators,”
*Journal of the London Mathematical Society*, vol. 66, no. 1, pp. 240–256, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2012 Thanyarat Jitpeera and Poom Kumam. 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.