1. Introduction

Throughout this paper, we always assume that is a real Hilbert space with inner product and norm , respectively, is a nonempty closed convex subset of , and is the metric projection of onto . In the following, we denote by strong convergence and by weak convergence. Recall that a mapping is called nonexpansive if

(1.1)

We denote by the set of fixed points of . Recall that a mapping is said to be

(i)monotone if , for all ;

(ii)-Lipschitz if there exists a constant such that , for all ;

(iii)-inverse-strongly monotone [1, 2] if there exists a positive real number such that

(1.2)

Remark 1.1.

It is obvious that any -inverse-strongly monotone mapping is monotone and -Lipschitz continuous.

Let be a mapping. The classical variational inequality problem is to find a such that

(1.3)

The set of solutions of variational inequality (1.3) is denoted by . The variational inequality has been extensively studied in the literature; see, for example, [3, 4] and the references therein.

A self-mapping is a contraction if there exists a constant such that

(1.4)

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [5–8] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert space:

(1.5)

where is a linear bounded operator, is the fixed point set of a nonexpansive mapping , and is a given point in . Let be a real Hilbert space. Recall that a linear bounded operator is strongly positive if there is a constant with property

(1.6)

Recently, Marino and Xu [9] introduced the following general iterative scheme based on the viscosity approximation method introduced by Moudafi [10]:

(1.7)

where is a strongly positive bounded linear operator on . They proved that if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (1.7) converges strongly to the unique solution of the variational inequality

(1.8)

which is the optimality condition for the minimization problem

(1.9)

where is a potential function for (i.e., for ).

On the other hand, two classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one is introduced by Mann [11] and is defined as follows:

(1.10)

where the sequence is in the interval .

The second iteration process is referred to as Ishikawa's iteration process [12] which is defined recursively by

(1.11)

where and are sequences in the interval . However, both (1.16) and (1.11) have only weak convergence in general (see [13], e.g.). Very recently, Qin and Cho [14] introduced a composite iterative algorithm defined as follows:

(1.12)

where is a contraction, is a nonexpansive mapping, and is a strongly positive linear bounded self-adjoint operator, proved that, under certain appropriate assumptions on the parameters, defined by (1.12) converges strongly to a fixed point of , which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).

On the other hand, for finding an element of , under the assumption that a set is nonempty, closed, and convex, a mapping is nonexpansive and a mapping is -inverse-strongly monotone, Takahashi and Toyoda [15] introduced the following iterative scheme:

(1.13)

where is a sequence in , and is a sequence in . They proved that if , then the sequence generated by (1.13) converges weakly to some . Recently, Iiduka and Takahashi [16] proposed another iterative scheme as follows

(1.14)

where is an -inverse strongly monotone mapping, and satisfy some parameters controlling conditions. They showed that if is nonempty, then the sequence generated by (1.14) converges strongly to some .

The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [17–20] and the references therein). The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings (see [21, 22]). The problem of finding an optimal point that minimizes a given cost function over the common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see [18, 23]). A simple algorithmic solution to the problem of minimizing a quadratic function over the common set of fixed points of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimation (see [23, 24]).

In this paper, we study the mapping defined by

(1.15)

where is a nonnegative real sequence with , for all , , form a family of infinitely nonexpansive mappings of into itself. Nonexpansivity of each ensures the nonexpansivity of . Such a is nonexpansive from to and it is called a -mapping generated by and .

In this paper, motivated and inspired by Su et al. [25], Marino and Xu [9], Takahashi and Toyoda [15], and Iiduka and Takahashi [16], we will introduce a new iterative scheme:

(1.16)

where is a mapping defined by (1.15), is a contraction, is strongly positive linear bounded self-adjoint operator, is a -inverse strongly monotone, and we prove that under certain appropriate assumptions on the sequences , , , and , the sequences defined by (1.16) converge strongly to a common element of the set of common fixed points of a family of and the set of solutions of the variational inequality for an inverse-strongly monotone mapping, which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).

2. Preliminaries

Let be a real Hilbert space. It is well known that for any

(2.1)

Let be a nonempty closed convex subset of . For every point , there exists a unique nearest point in , denoted by , such that

(2.2)

is called the metric projection of onto . It is well known that is a nonexpansive mapping of onto and satisfies

(2.3)

for every Moreover, is characterized by the following properties: and

(2.4)

for all . It is easy to see that the following is true:

(2.5)

A Banach space is said to satisfy the Opial's condition if for each sequence in which converges weakly to a point we have

(2.6)

It is well known that each Hilbert space satisfies the Opial's condition.

A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if the graph of of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies . Let be a monotone map of into and let be the normal cone to at , that is, and define

(2.7)

Then is the maximal monotone and if and only if ; see [26].

Now we collect some useful lemmas for proving the convergence result of this paper.

Lemma 2.1.

In a Hilbert space . Then the following inequality holds

(2.8)

Lemma 2.2 (see [27]).

Let and be bounded sequences in a Banach space and let be a sequence in with Suppose for all integers and Then,

Lemma 2.3 (see [28]).

Assume is a sequence of nonnegative real numbers such that

(2.9)

where is a sequence in and is a sequence in such that

(1)

(2) or

Then

Lemma 2.4 (see [9]).

Assume that is a strongly positive linear bounded self-adjoint operator on a Hilbert space with coefficient and . Then .

Throughout this paper, we will assume that , for all . Concerning defined by (1.15), we have the following lemmas which are important to prove our main result.

Lemma 2.5 (see [29]).

Let be a nonempty closed convex subset of a Hilbert space , let be a family of infinitely nonexpansive mapping with , and let be a real sequence such that , for all . Then

(1) is nonexpansive and for each ;

(2)for each and for each positive integer , the limit exists;

(3)the mapping define by

(2.10)

is a nonexpansive mapping satisfying and it is called the -mapping generated by and

Lemma 2.6 (see [30]).

Let be a nonempty closed convex subset of a Hilbert space , let be a family of infinitely nonexpansive mappings with , and let be a real sequence such that , for all . If is any bounded subset of , then

(2.11)

3. Main Results

Now we are in a position to state and prove the main result in this paper.

Theorem 3.1.

Let be a closed convex subset of a real Hilbert space , let be a contraction of into itself, let be an -inverse strongly monotone mapping of into , and let be a family of infinitely nonexpansive mappings with . Let be a strongly positive linear bounded self-adjoint operator with the coefficient such that . Assume that . Let , , , and be sequences in satisfying the following conditions:

(C1)

(C2)

(C3)

(C4)

(C5).

Then the sequence defined by (1.16) converges strongly to , where which solves the following variational inequality:

(3.1)

Proof.

Since as by the condition (C1), we may assume, without loss of generality that for all . First, we will show that is nonexpansive. Indeed, for all and ,

(3.2)

which implies that is nonexpansive. Noticing that is a linear bounded self-adjoint operator, one has

(3.3)

Observing that

we obtain is positive. It follows that

Next, we observe that is bounded. Indeed, pick and notice that

(3.4)

It follows that

(3.5)

By simple induction, we have

(3.6)

which gives that the sequence is bounded, and so are and .

Next, we claim that

(3.7)

Since and are nonexpansive, we have

(3.8)

where is a constant such that . Similarly, there exists such that .

Observing that

(3.9)

we obtain that

(3.10)

It follows that

(3.11)

Noticing that

(3.12)

we obtain

(3.13)

It follows that

(3.14)

Substituting (3.11) into (3.14), we get

(3.15)

where is an appropriate constant such that

(3.16)

Putting , we get, .

Now, we compute . Observing that

(3.17)

It follows from (3.15) that

(3.18)

It follows that

(3.19)

Observing the conditions (C1) and (C4) and taking the superior limit as , we get

(3.20)

We can obtain easily by Lemma 2.2 since

(3.21)

one obtains that (3.7) holds. Setting , we have

(3.22)

Observing that

(3.23)

we arrive at

(3.24)

This implies

(3.25)

From (3.7) and (C1) we obtain that

(3.26)

Next, we will show that as for any Observe that

(3.27)

where

(3.28)

This impies that

(3.29)

Since and from (3.7), we obtain

(3.30)

From (2.3), we have

(3.31)

so, we obtain

(3.32)

It follows that

(3.33)

which implies that

(3.34)

Applying (3.7), (3.30), and to the last inequality, we obtain that

(3.35)

It follows from (3.26) and (3.35) that

(3.36)

On the other hand, one has

(3.37)

which implies

(3.38)

From the conditions (C3), it follows that

(3.39)

Applying Lemma 2.6 and (3.39), we obtain that

(3.40)

It follows from (3.26) and (3.40) that

(3.41)

We observe that is a contraction. Indeed, for all , we have

(3.42)

Banach's Contraction Mapping Principle guarantees that has a unique fixed point, say . That is, .

Next, we claim that

(3.43)

Indeed, we choose a subsequence of such that

(3.44)

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . From we obtain Therefore, we have

(3.45)

Next we prove that .

First, we prove that .

Suppose the contrary, , that is, . Since , by the Opial's condition and (3.41), we have

(3.46)

This is a contradiction, which shows that .

Next, we prove . For this purpose, let be the maximal monotone mapping defined by (2.7):

(3.47)

For any given , hence . Since we have

(3.48)

On the other hand, from , we have

(3.49)

that is,

(3.50)

Therefore, we obtian

(3.51)

Noting that as and is Lipschitz continuous, hence from (3.18), we obtain

(3.52)

Since is maximal monotone, we have , and hence .

The conclusion is proved.

Hence by (3.45), we obtain

(3.53)

Since , it follows from (3.39), (3.41), and (3.53) that

(3.54)

Hence (3.43) holds. Using (3.26) and (3.54), we have

(3.55)

Now, from Lemma 2.1, it follows that

(3.56)

Since , , and are bounded, we can take a constant such that

(3.57)

for all . It then follows that

(3.58)

where

(3.59)

Using (C1), (3.54), and (3.55), we get . Now applying Lemma 2.3 to (3.58), we conclude that . This completes the proof.

Remark 3.2.

Theorem 3.1 mainly improve the results of Qin and Cho [14] from a single nonexpansive mapping to an infinite family of nonexpansive mappings.

4. Applications

In this section, we obtain two results by using a special case of the proposed method.

Theorem 4.1.

Let be a real Hilbert space, let be an -inverse strongly monotone mapping on , let be a family of infinitely nonexpansive mappings with . Let a contraction with coefficient , and let be a strongly positive bounded linear operator on with coefficient and . Suppose the sequences , , and be generated by

(4.1)

where , , , and are sequences in satisfying the following conditions:

(C1)

(C2)

(C3)

(C4)

(C5).

Then , , and converge strongly to which solves the variational inequality:

(4.2)

Proof.

We have and . Applying Theorem 3.1, we obtain the desired result.

Next, we will apply the main results to the problem for finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of -strictly pseudocontractive mappings.

Definition 4.2.

A mappings is said to be a -strictly pseudocontractive mapping if there exists such that

(4.3)

The following lemmas can be obtained from [31, Proposition 2.6] by Acedo and Xu, easily.

Lemma 4.3.

Let be a Hilbert space, let be a closed convex subset of . For any integer , assume that, for each is a -strictly pseudocontractive mapping for some . Assume that is a positive sequence such that . Then is a -strictly pseudocontractive mapping with .

Lemma 4.4.

Let and be as in Lemma 4.3. Suppose that has a common fixed point in . Then .

Let be a -strictly pseudocontractive mapping for some . We define a mapping , where is a positive sequence such that , then is a -inverse-strongly monotone mapping with . In fact, from Lemma 4.3, we have

(4.4)

That is,

(4.5)

On the other hand

(4.6)

Hence, we have

(4.7)

This shows that is -inverse-strongly monotone.

Theorem 4.5.

Let be a closed convex subset of a real Hilbert space . For any integer , assume that, for each is a -strictly pseudocontractive mapping for some . Let be a family of infinitely nonexpansive mappings with . Let a contraction with coefficient and let be a strongly positive bounded linear operator with coefficient and . Let the sequences , , and be generated by

(4.8)

where , , , and are the sequences in satisfying the following conditions:

(C1)

(C2)

(C3)

(C4)

(C5)

Then , , and converge strongly to which solves the variational inequality:

(4.9)

Proof.

Taking , we know that is -inverse strongly monotone with . Hence, is a monotone -Lipschitz continuous mapping with . From Lemma 4.4, we know that is a -strictly pseudocontractive mapping with and then by Chang [30, Proposition 1.3.5]. Observe that

(4.10)

The conclusion of Theorem 4.5 can be obtained from Theorem 3.1.

Remark 4.6.

Theorem 4.5 is a generalization and improvement of the theorems by Qin and Cho [14], Iiduka and Takahashi [16, Thorem 3.1], and Takahashi and Toyoda [15].