Research Article | Open Access

# Shrinking Projection Method for Fixed Point Problems of an Infinite Family of Strictly Pseudocontractive Mappings and the System of Cocoercive Quasivariational Inclusions Problems in Hilbert Spaces

**Academic Editor:**C. Zhu

#### Abstract

This paper is concerned with a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

#### 1. Introduction

Throughout this paper, we always assume that is a nonempty closed convex subset of a real Hilbert space with inner product and norm denoted by and , respectively, denoting the family of all the nonempty subsets of .

Let be a single-valued nonlinear mapping and a set-valued mapping. We consider the following * quasivariational inclusion problem*, which is to find a point ,
where is the zero vector in . The set of solutions of problem (1.1) is denoted by .

Recall that is the * metric projection* of onto ; that is, for each , there exists the unique point in such that . A mapping is called * nonexpansive* if for all . A point is a * fixed point* of provided . We denote by the set of fixed points of ; that is, . If is nonempty bounded closed convex subset of and is a nonexpansive mapping of into itself, then is nonempty (see [1]). Recall that a mapping is said to be (i)*monotone* if
(ii)*-Lipschitz continuous* if there exists a constant such that
if , then is a nonexpansive, (iii)*pseudocontractive* if
(iv)*-strictly pseudocontractive* if there exists a constant such that
it is obvious that is a nonexpansive if and only if is 0-strictly pseudocontractive, (v)*-strongly monotone* if there exists a constant such that
(vi)*-inverse-strongly monotone (or **-cocoercive)* if there exists a constant such that
if , then is said to be *firmly nonexpansive*; it is obvious that any -inverse-strongly monotone mapping is monotone and (1/)-Lipschitz continuous.

The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [2–5] and the references therein).

In this paper, we study the mapping defined by
where is nonnegative real sequence in , for all , from a family of infinitely nonexpansive mappings of into itself. It is obvious that is a nonexpansive mapping of into itself; such a mapping is called a *-mapping* generated by and .

*Definition 1.1 (see [6]). *Let be a multivalued maximal monotone mapping. Then, the single-valued mapping defined by , for all , is called * the resolvent operator associated with *, where is any positive number and is the identity mapping.

Recently, Zhang et al. [6] considered the problem (1.1) and the problem of a fixed point of nonexpansive mapping. To be more precise, they proved the following theorem.

Theorem ZLC. *Let be a real Hilbert space, an -inverse-strongly monotone mapping, a maximal monotone mapping, and a nonexpansive mapping. Suppose that the set , where is the set of solutions of quasivariational inclusion (1.1). Suppose that and is the sequence defined by
**
for all , where and satisfying the following conditions: **(C1) and , **(C2) . **
Then, converges strongly to .*

Nakajo and Takahashi [7] introduced an iterative scheme for finding a fixed point of a nonexpansive mapping by a hybrid method which is called that shrinking projection method (or method) as in the following theorem.

Theorem NT. *Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive mapping of into itself such that . Suppose that and is the sequence defined by
**
where . Then, converges strongly to . *

In the same way, Kikkawa and Takahashi [8] introduced an iterative scheme for finding a common fixed point of an infinite family of nonexpansive mappings as follows: where and is a -mapping of into itself generated by and . They prove that, if , then the sequence generated by (1.11) converges strongly to .

Recently, Su and Qin [9] modified the shrinking projection method for finding a fixed point of a nonexpansive mapping, for which the convergence rate of the iterative scheme is faster than that of the iterative scheme of Nakajo and Takahashi [7] as follows: where and is a nonexpansive mapping of into itself. They prove that, under the parameter , if , then the sequence generated by (1.12) converges strongly to .

On the other hand, Tada and Takahashi [10] introduced an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of solutions of a fixed point problem of a nonexpansive mapping as follows:
where , is a nonexpansive mapping of into and is a bifunction from into . They prove that, under the sequences for some and for some , if , then the sequence generated by (1.13) converges strongly to such that is the set of solutions of * equilibrium problem* defined by

In this paper, we introduce an iterative scheme (1.15) for finding a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems by the shrinking projection method in Hilbert spaces as follows: where chosen arbitrarily, is a maximal monotone mapping, is a -cocoercive mapping for each , and is a -mapping on generated by and such that the mapping defined by for all , where is an infinite family of -strictly pseudocontractive mappings with a fixed point.

It is well known that the class of strictly pseudocontractive mappings contains the class of nonexpansive mappings, and it follows that, if , then the iterative scheme (1.15) is reduced to find a common element of the set of common fixed points for an infinite family of nonexpansive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces.

Furthermore, if for all and , then the iterative scheme (1.15) is reduced to extend and improve the results of Kikkawa and Takahashi [8] for finding a common fixed point of an infinite family of -strictly pseudocontractive mappings as follows: and if and setting , for all , then the iterative scheme (1.16) is reduced to find a fixed point of a nonexpansive mapping, for which the convergence rate of the iterative scheme is faster than that of the iterative scheme of Su and Qin [9] as follows:

We suggest and analyze the iterative scheme (1.15) under some appropriate conditions imposed on the parameters. The strong convergence theorem for the above two sets is obtained, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

#### 2. Preliminaries

We collect the following lemmas which will be used in the proof of the main results in the next section.

Lemma 2.1 (see [11]). *Let be a Hilbert space. For any and , one has
*

Lemma 2.2 (see [1]). *Let be a nonempty closed convex subset of a Hilbert space . Then the following inequality holds:
*

Lemma 2.3 (see [5]). *Let be a nonempty closed convex subset of a Hilbert space , define mapping as (1.8), let be a family of infinitely nonexpansive mappings with , and let be a sequence such that , for all . Then *

(1)* is nonexpansive and for each , *(2)*for each and for each positive integer , exists, *(3)*the mapping defined by
is a nonexpansive mapping satisfying and it is called the -mapping generated by and . *

Lemma 2.4 (see [6]). *The resolvent operator associated with is single valued and nonexpansive for all .*

Lemma 2.5 (see [6]). * is a solution of quasivariational inclusion (1.1) if and only if , for all , that is,
*

Lemma 2.6 (see [12]). *Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose that . Let be a sequence of positive real numbers such that . Then a mapping on defined by
**
for , is well defined, nonexpansive and holds.*

Lemma 2.7 (see [13]). *Let be a nonempty closed convex subset of a Hilbert space and a nonexpansive mapping. Then is demiclosed at zero. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that .*

Lemma 2.8 (see [14]). *Let be a nonempty closed convex subset of a real Hilbert space and a -strict pseudocontraction. Define by for each . Then, as , S is nonexpansive such that .*

Lemma 2.9 (see [1]). *Every Hilbert space has Radon-Riesz property or Kadec-Klee property, that is, for a sequence with and then .*

#### 3. Main Results

Theorem 3.1. *Let be a real Hilbert space, a maximal monotone mapping, and a -cocoercive mapping for each . Let be an infinite family of -strictly pseudocontractive mappings with a fixed point such that . Define a mapping by
**
for all , where . Let be a -mapping generated by and such that , for some . Assume that . For chosen arbitrarily, suppose that is generated iteratively by
**
where **(C1) such that , **(C2) and for each , **(C3) . **
Then, the sequences and converge strongly to .*

*Proof. *For any and for each , by the -cocoercivity of , we have
which implies that is nonexpansive. Pick . Therefore, by Lemma 2.5, we have
for each . Since , where and is a family of -strict pseudocontraction, therefore, by Lemma 2.8, we have that is nonexpansive and . It follows from Lemma 2.3(1) that , which implies that . Therefore, by (C3), (3.4), Lemma 2.1, and the nonexpansiveness of , and , we have
for all . Firstly, we prove that is closed and convex for all . It is obvious that is closed and, by mathematical induction, that is closed for all , that is is closed for all . Since is equivalent to
for all , therefore, for any and , we have
for all , and we have
for all . Since is convex, and by putting in (3.6), (3.7), and (3.8), we have that is convex. Suppose that is given and is convex for some . It follows by putting in (3.6), (3.7), and (3.8) that is convex. Therefore, by mathematical induction, we have that is convex for all , that is, is convex for all . Hence, we obtain that is closed and convex for all .

Next, we prove that for all . It is obvious that . Therefore, by (3.2) and (3.5), we have and note that , and so . Hence, we have . Since is a nonempty closed convex subset of , there exists a unique element such that . Suppose that is given such that , and for some . Therefore, by (3.2) and (3.5), we have . Since , therefore, by Lemma 2.2, we have
for all . Thus, by (3.2), we have , and so . Hence, we have . Since is a nonempty closed convex subset of , there exists a unique element such that . Therefore, by mathematical induction, we obtain for all , and so for all , and we can define for all . Hence, we obtain that the iteration (3.2) is well defined.

Next, we prove that is bounded. Since for all , we have
for all . It follows by that for all . This implies that is bounded, and so is .

Next, we prove that as . Since , therefore, by (3.10), we have for all . This implies that is a bounded nondecreasing sequence and there exists the limit of , that is,
for some . Since , therefore, by (3.2), we have
It follows by (3.12) that
Therefore, by (3.11), we have
Since , therefore, by (3.2), we have
It follows by (3.15) that
Therefore, by (3.14), we obtain

Since is bounded, there exists a subsequence of which converges weakly to . Next, we prove that . Define the sequence of mappings and the mapping by
for all . Therefore, by (C1) and Lemma 2.3(3), we have
where . From (C3) and Lemma 2.3(3), we have that and are nonexpansive. Therefore, by (C2), (C3), Lemmas 2.3(3), 2.5, 2.6, and 2.8, we have
that is, . From (3.17), we have . Thus, from (3.2) and (3.18), we get . It follows from and Lemma 2.7 that , that is, .

Since is a nonempty closed convex subset of , there exists a unique such that . Next, we prove that as . Since , we have for all , and it follows that
Since , therefore, by (3.10), we have
Therefore, by (3.21), (3.22), and the weak lower semicontinuity of norm, we have
It follows that
Since as , therefore, we have
Hence, from (3.24), (3.25), the Kadec-Klee property, and the uniqueness of , we obtain
It follows that converges strongly to , and so is . This completes the proof.

*Remark 3.2. *The iteration (3.2) is the difference with some well known results as the following.

(1)The sequence is the projection sequence of onto for all such that
(2)The proof of is simple by the demiclosedness principle because the sequence is a linear nonexpansive mapping form of the mappings and .(3)Solving a common fixed point for an infinite family of strictly pseudocontractive mappings and a system of cocoercive quasivariational inclusions problems by iteration is obtained.

#### 4. Applications

Theorem 4.1. *Let be a real Hilbert space, a maximal monotone mapping, and a -cocoercive mapping for each . Let be an infinite family of nonexpansive mappings. Define a mapping by
**
for all , where . Let be a -mapping generated by and such that , for some . Assume that . For chosen arbitrarily, suppose that is generated iteratively by
**
where **(C1) such that , **(C2) and for each , **(C3) . **
Then the sequences and converge strongly to .*

*Proof. *It is concluded from Theorem 3.1 immediately, by putting .

Theorem 4.2. *Let be a nonempty closed convex subset of a real Hilbert space and an infinite family of -strictly pseudocontractive mappings with a fixed point such that . Define a mapping by
**
for all , where . Let be a -mapping generated by and such that , for some . Assume that . For chosen arbitrarily, suppose that is generated iteratively by
**
where such that . Then the sequences and converge strongly to .*

*Proof. *It is concluded from Theorem 3.1 immediately, by putting for all .

Theorem 4.3. *Let be a nonempty closed convex subset of a real Hilbert space and a nonexpansive mapping. Assume that . For chosen arbitrarily, suppose that is generated iteratively by
**
where such that . Then the sequences and converge strongly to .*

*Proof. *It is concluded from Theorem 4.2, by putting . Setting , for all and leting for some , therefore, from the definition of in Theorem 4.2, we have and for all . Since is a -mapping generated by and , therefore, by the definition of and in (1.8), we have for all and . Hence, by Theorem 4.2, we obtain
where . Since, the same as in the proof of Theorem 3.1, we have that is a nonempty closed convex subset of for all and by Theorem 4.2, we have
for all . It follows that for all , where is defined as in Theorem 4.2. Hence, by Theorem 4.2, we obtain the desired result. This completes the proof.

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#### Copyright

Copyright © 2011 Pattanapong Tianchai. 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.