## Existence and Uniqueness of Fixed Point in Various Abstract Spaces and Related Applications

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Poom Kumam, Thanyarat Jitpeera, "Strong Convergence of an Iterative Algorithm for Hierarchical Problems", *Abstract and Applied Analysis*, vol. 2014, Article ID 678147, 9 pages, 2014. https://doi.org/10.1155/2014/678147

# Strong Convergence of an Iterative Algorithm for Hierarchical Problems

**Academic Editor:**Wei-Shih Du

#### Abstract

We introduce the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence of the algorithm is proved under some mild conditions. Our results extend those of Yao et al., Iiduka, Ceng et al., and other authors.

#### 1. Introduction

Let be a closed convex subset of a real Hilbert space with inner product and norm . We denote weak convergence and strong convergence by notations and , respectively. Let be a nonlinear mapping. The* Hartman-Stampacchia variational inequality* [1] is to find such that . The set of solutions is denoted by . is said to be a -*contraction* if there exists a constant such that . A mapping is said to be* monotone* if . A mapping is said to be -* strongly monotone* if there exists a positive real number such that . A mapping is said to be -*inverse-strongly monotone* if there exists a positive real number such that . A mapping is said to be -*Lipschitz continuous* if there exists a positive real number such that . A linear bounded operator is said to be* strongly positive* on if there exists a constant with the property . A mapping is said to be* nonexpansive* if .

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 [2]).

We discuss the following variational inequality problem over the fixed point set of a nonexpansive mapping (see [3–16]), which is said to be the* hierarchical problem*. Let a monotone, continuous mapping and a nonexpansive mapping . Find , where . This solution set is denoted by .

We introduce the following variational inequality problem over the solution set of variational inequality problem and the fixed point set of a nonexpansive mapping (see [17, 18]), which is said to be the* triple hierarchical problem*. Let an inverse-strongly monotone , a strongly monotone and Lipschitz continuous , and a nonexpansive mapping . Find , where .

In 2009, Yao et al. [19] considered the following two-step iterative algorithm with the initial guess which is chosen arbitrarily: where satisfies certain assumptions. Let be two nonexpansive mappings and let be a contraction mapping. Then, they proved that the above iterative sequence converges strongly to fixed point.

Next, Iiduka [17] introduced a monotone variational inequality with variational inequality constraint over the fixed point set of a nonexpansive mapping; the sequence defined by the iterative method below, with the initial guess , is chosen arbitrarily: where and satisfy certain conditions, is an inverse-strongly monotone, is a strongly monotone and Lipschitz continuous, and is a nonexpansive mapping; then the strongly convergence analysis of the sequence generated by (2) is proved under some appropriate conditions.

In 2011, Yao et al. [20] studied the hierarchical problem over the fixed point set. Let the sequences be generated by these two following algorithms: implicit algorithm explicit algorithm .They illustrated that these two algorithms converge strongly to the unique solution of the variational inequality which is to find such that where is a strongly positive linear bounded operator, is a -contraction, and is a nonexpansive mapping satisfying some conditions.

Very recently, Ceng et al. [21] studied the following new algorithms. For is chosen arbitrarily, they defined a sequence by where the mappings , are nonexpansive mappings with . Let be a Lipschitzian and strongly monotone operator and let be a contraction mapping satisfying some appropriate conditions. They proved that the proposed algorithms strongly converge to the minimum norm fixed point of .

In this paper, we consider a new iterative algorithm for solving the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping which contain algorithms (1) and (4) as follows: where the mappings , are nonexpansive mappings with . Let be a Lipschitzian and strongly monotone operator, and let be a contraction mapping satisfying some mild conditions. Find a point such that This solution set of (6) is denoted by . The strong convergence for the proposed algorithms to the solution is solved under some appropriate assumptions. Our results improve the results of Ceng et al. [21], Iiduka [17], Yao et al. [19], Yao et al. [20], and some authors.

#### 2. Preliminaries

Let be a nonempty closed convex subset of . There holds the following inequality in an inner product space . For every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto . It is well known that is a nonexpansive mapping of onto and satisfies for every . Moreover, is characterized by the following properties: and for all . Let be a monotone mapping of into . In the context of the variational inequality problem the characterization of projection (9) implies the following: It is also known that satisfies the Opial’s condition [22]; that is, for any sequence with , the inequality holds for every with .

Lemma 1 (see [23]). *Let be a closed convex subset of a real Hilbert space and let be a nonexpansive mapping. Then is demiclosed at zero; that is, and imply .*

Lemma 2 (see [24]). *Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .*

Lemma 3 (see [10]). *Let be -strongly monotone and -Lipschitz continuous and let . For , define by for all . Then, for all , hold, where .*

Lemma 4 (see [25]). *Assume that is a sequence of nonnegative real numbers such that
**
where and is a sequence in such that*(i)*;*(ii)* or .**Then .*

#### 3. Strong Convergence Theorem

In this section, we introduce an iterative algorithm of triple hierarchical for solving monotone variational inequality problems for -Lipschitzian and -strongly monotone operators over the solution set of variational inequality problems and the fixed point set of a nonexpansive mapping.

Theorem 5. *Let be a nonempty closed and convex subset of a real Hilbert space . Let be -Lipschitzian and -strongly monotone operators with constant and , respectively, and let be a -contraction with coefficient . Let be a nonexpansive mapping with , and let be a nonexpansive mapping. Let and , where . Suppose that is a sequence generated by the following algorithm where is chosen arbitrarily:
**
where satisfy the following conditions:*(C1):*;*(C2):*, , ;*(C3):*.**Then converges strongly to , which is the unique solution of another variational inequality:
**
where .*

*Proof. *We will divide the proof into four steps.*Step 1*. We will show that is bounded. Indeed, for any , we have
From (13), we deduce that
Substituting (15) into (16), we obtain
By induction, it follows that
Therefore, is bounded and so are , , , , and .*Step 2*. We will show that . Setting , we obtain
which implies that
It follows from (13) that
where is a constant such that
Hence, conditions (C2) and (C3) allow us to apply Lemma 4; then we get
By (21), we get
Using the conditions (C2) and (C3), we can apply Lemma 4 to conclude that
By (13), we compute
From the condition (C2), we note that . At the same time, from (13), we also have
By the conditions (C1) and (C2), we note that . Consider
From (23), (26), and (27), we obtain
We set ; then we get
From (13), we have
By the conditions (C1) and (C2) again, we note that . Consider
From (29), , and , we obtain
*Step 3*. We will show that . Rewrite (13) as
We observe that
Set
We note from (35) that
This yields that, for each ,
In view of (38), is nonnegative due to the monotonicity of . From (38), we derive that
Since (29) implies , as , from (25), then we get . Using (C1) and (30), , as and is bounded. We obtain from (39) that
Since the sequence is bounded, we can take a subsequence of such that
and . From (33), by the demiclosed principle of the nonexpansive mapping, it follows that . Then
*Step 4*. Finally, we will prove . From (13), we note that
Using (43), we compute
Since , , and are all bounded, we can choose a constant such that
It follows that
where
Now, applying Lemma 4 and (35), we conclude that . This completes the proof.

Corollary 6. *Let be a nonempty closed and convex subset of a real Hilbert space . Let be -Lipschitzian and -strongly monotone operators with constant and , respectively. Let be a nonexpansive mapping with , and let be a nonexpansive mapping. Let and , where . Suppose is a sequence generated by the following algorithm arbitrarily:
**
where satisfy the following conditions (C1)–(C3). Then converges strongly to , which is the unique solution of variational inequality:
**
where .*

*Proof. *Putting in Theorem 5, we can obtain the desired conclusion immediately.

Corollary 7. *Let be a nonempty closed and convex subset of a real Hilbert space . Let be a -contraction with coefficient , and let be a nonexpansive mapping with and a nonexpansive mapping. Suppose is a sequence generated by the following algorithm, , arbitrarily:
**
where satisfy the following conditions (C1)–(C3). Then converges strongly to , which is the unique solution of variational inequality:
**
where .*

*Proof. *Putting , , and in Theorem 5, we can obtain the desired conclusion immediately.

Corollary 8. *Let be a nonempty closed and convex subset of a real Hilbert space . Let be a nonexpansive mapping with and let be a nonexpansive mapping. Suppose is a sequence generated by the following algorithm, , arbitrarily:
**
where satisfy the following conditions (C1)–(C3). Then converges strongly to , which is the unique solution of variational inequality:
*

*Proof. *Putting in Corollary 7, we can obtain the desired conclusion immediately.

Corollary 9. *Let be a nonempty closed and convex subset of a real Hilbert space . Let be a -contraction with coefficient , and let be a nonexpansive mapping with and a nonexpansive mapping. Suppose is a sequence generated by the following algorithm, , arbitrarily:
*