New Iterative Scheme for Finite Families of Equilibrium, Variational Inequality, and Fixed Point Problems in Banach Spaces

Abstract

We introduced a new iterative scheme for finding a common element in the set of common fixed points of a finite family of quasi-ϕ-nonexpansive mappings, the set of common solutions of a finite family of equilibrium problems, and the set of common solutions of a finite family of variational inequality problems in Banach spaces. The proof method for the main result is simplified under some new assumptions on the bifunctions.

1. Introduction

Throughout this paper, let denote the set of all real numbers. Let be a smooth Banach space and the dual space of . The function is defined by

(1.1)

where is the normalized dual mapping from to defined by

(1.2)

Let be a nonempty closed and convex subset of . The generalized projection is a mapping that assigns to an arbitrary point the minimum point of the function , that is, , where is the solution to the minimization problem

(1.3)

In Hilbert spaces, and , where is the metric projection. It is obvious from the definition of function that

(1.4)

We remark that if is a reflexive, strictly convex and smooth Banach space, then for , if and only if . For more details on and , the readers are referred to [1–4].

Let be a mapping from into itself. We denote the set of fixed points of by . is called to be nonexpansive if for all and quasi-nonexpansive if and for all and . A point is called to be an asymptotic fixed point of [5] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of is denoted by . The mapping is said to be relatively nonexpansive [6–8] if and for all and . The mapping is said to be -nonexpansive if for all . is called to be quasi--nonexpansive [9] if and for all and .

In 2005, Matsushita and Takahashi [10] introduced the following algorithm:

(1.5)

where is the duality mapping on , is a relatively nonexpansive mapping from into itself, and is a sequence of real numbers such that and and proved that the sequence generated by (1.5) converges strongly to , where is the generalized projection from onto .

Let be a bifunction from to . The equilibrium problem for is to find such that

(1.6)

We use to denote the solution set of the equilibrium problem (1.6). That is,

(1.7)

For studying the equilibrium problem, is usually assumed to satisfy the following conditions:

(A1) for all ;

(A2) is monotone, that is, for all ;

(A3)for each , ;

(A4)for each , is convex and lower semicontinuous.

Recently, many authors investigated the equilibrium problems in Hilbert spaces or Banach spaces; see, for example, [11–25]. In [20], Qin et al. considered the following iterative scheme by a hybrid method in a Banach space:

(1.8)

where is a closed quasi--nonexpansive mapping for each , are real sequences in satisfying for each and for each and is a real sequence in with . Then the authors proved that converges strongly to , where .

Very recently, Zegeye and Shahzad [25] introduced a new scheme for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions set of finite family of equilibrium problems, and common solutions set of finite family of variational inequality problems for monotone mappings in a Banach space. More precisely, let , , be a finite family of bifunctions, , , a finite family of relatively quasi-nonexpansive mappings, and , , a finite family of continuous monotone mappings. For , define the mappings , by

(1.9)

where and for some . Zegeye and Shahzad [25] introduced the following scheme:

(1.10)

where , such that . Further, they proved that converges strongly to an element of , where .

In this paper, motivated and inspired by the iterations (1.8) and (1.10), we consider a new iterative process with a finite family of quasi--nonexpansive mappings for a finite family of equilibrium problems and a finite family of variational inequality problems in a Banach space. More precisely, let be a family of quasi--nonexpansive mappings, a finite family of bifunctions, and a finite family of continuous monotone mappings such that . Let and . Define the mappings , by

(1.11)

(1.12)

Consider the iteration

(1.13)

where are the real numbers in satisfying and for each , are the real numbers in satisfying . We will prove that the sequence generated by (1.13) converges strongly to an element in . In this paper, in order to simplify the proof, we will replace the condition (A3) with (A3'): for each fixed , is continuous.

Obviously, the condition (A3') implies (A3). Under the condition (A3'), we will show that each (as well as , , ) is closed which is such that the proof for the main result of this paper is simplified.

2. Preliminaries

The modulus of smoothness of a Banach space is the function defined by

(2.1)

The space is said to be smooth if , , and is called uniformly smooth if and only if .

A Banach space is said to be strictly convex if for all with and . It is said to be uniformly convex if for any two sequences and in such that and . It is known that if a Banach space is uniformly smooth, then its dual space is uniformly convex.

A Banach space is called to have the Kadec-Klee property if for any sequence and with , where denotes the weak convergence, and , then as , where denotes the strong convergence. It is well known that every uniformly convex Banach space has the Kadec-Klee property. For more details on the Kadec-Klee property, the reader is referred to [3, 4].

Let be a nonempty closed and convex subset of a Banach space . A mapping is said to be closed if for any sequence such that and , .

Let be a mapping. is said to be monotone if for each , the following inequality holds:

(2.2)

Let be a monotone mapping from into . The variational inequality problem on is formulated as follows:

(2.3)

The solution set of the above variational inequality problem is denoted by .

Next we state some lemmas which will be used later.

Lemma 2.1 (see [1]).

Let be a nonempty closed and convex subset of a smooth Banach space and . Then, if and only if

(2.4)

Lemma 2.2 (see [1]).

Let be a reflexive, strictly convex and smooth Banach space, a nonempty closed and convex subset of , and . Then

(2.5)

Lemma 2.3 (see [20]).

Let be a strictly convex and smooth Banach space, a nonempty closed and convex subset of , and a quasi--nonexpansive mapping. Then is a closed and convex subset of .

Since the condition (A3') implies (A3), the following lemma is a natural result of [22, Lemmas  2.8 and 2.9].

Lemma 2.4.

Let be a closed and convex subset of a smooth, strictly convex and reflexive Banach space . Let be a bifunction from satisfying (A1), (A2), (A3'), and (A4). Let and . Then

• (a) there exists such that

(2.6)

• (b) define a mapping by

(2.7)

Then the following conclusions hold:

(1) is single-valued;

(2) is firmly nonexpansive, that is, for all ,

(2.8)

(3);

(4) is quasi--nonexpansive;

(5) is closed and convex;

(6).

Remark 2.5.

Let be a continuous monotone mapping and define for all . It is easy to see that satisfies the conditions (A1), (A2), (A3'), and (A4) and . Hence, for every real number , if defining a mapping by

(2.9)

then satisfies all the conclusions in Lemma 2.4. See [25, Lemma 2.4].

Lemma 2.6 (see [26]).

Let and be two fixed real numbers. Then a Banach space is uniformly convex if and only if there exists a continuous strictly increasing convex function with such that

(2.10)

for all and , where .

The following lemma can be obtained from Lemma 2.6 immediately; also see [20, Lemma  1.9].

Lemma 2.7 (see [20]).

Let be a uniformly convex Banach space, a positive number, and a closed ball of . There exists a continuous, strictly increasing and convex function with such that

(2.11)

for all and such that .

Lemma 2.8.

Let be a closed and convex subset of a uniformly smooth and strictly convex Banach space . Let be a bifunction satisfying (A1), (A2), (A3'), and (A4). Let and be a mapping defined by (2.7). Then T_r is closed.

Proof.

Let converge to and converge to . To end the conclusion, we need to prove that . Indeed, for each , Lemma 2.4 shows that there exists a unique such that , that is,

(2.12)

Since is uniformly smooth, is continuous on bounded set (note that and are both bounded). Taking the limit as in (2.12), by using (A3'), we get

(2.13)

which implies that . This completes the proof.

3. Main Results

Theorem 3.1.

Let be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property. Let be a family of closed quasi--nonexpansive mappings, a finite family of bifunctions satisfying the conditions (A1), (A2), (A3'), and (A4), and a finite family of continuous monotone mappings such that . Let . Let be a sequence generated by the following manner:

(3.1)

where and are defined by (1.11) and (1.12), are the real numbers in satisfying and for each , are the real numbers in satisfying . Then the sequence converges strongly to , where is the generalized projection from onto .

Proof.

First we prove that is closed and convex for each . From the definition of , it is obvious that is closed. Moreover, since is equivalent to , it follows that is convex for each . By the definition of , we can conclude that is closed and convex for each .

Next, we prove that for each . From Lemma 2.4 and Remark 2.5, we see that each () and () are quasi--nonexpansive. Hence, for any , we have

(3.2)

which implies that for each . So, it follows from the definition of that for each . Therefore, the sequence is well defined. Also, from Lemma 2.2 we see that

(3.3)

for each . This shows that the sequence is bounded. It follows from (1.4) that the sequence is also bounded.

Since is reflexive, we may, without loss of generality, assume that . Since is closed and convex for each , we can conclude that for each . By the definition of , we see that

(3.4)

It follows that

(3.5)

This implies that

(3.6)

Hence, we have as . In view of the Kadec-Klee property of , we get that

(3.7)

By the construction of , we have that and . It follows from Lemma 2.2 that

(3.8)

Letting , we obtain that . In view of , we have and hence

(3.9)

It follows that

(3.10)

From (1.4), we see that

(3.11)

Hence,

(3.12)

This implies that the sequence is bounded. Note that reflexivity of implies reflexivity of . Thus, we may assume that . Furthermore, reflexivity of implies that there exists such that . Then, it follows that

(3.13)

Take on both sides of (3.13) over and use weak lower semicontinuity of norm to get that

(3.14)

which implies that . Hence, . It follows that . Now, from (3.12) and Kadec-Klee property of , we obtain that as . Then the demicontinuity of implies that . Now, from (3.11) and the fact that has the Kadec-Klee property, we obtain that . Note that

(3.15)

It follows that

(3.16)

Since is uniformly norm-to-norm continuous on any bounded sets, we have

(3.17)

Since is uniformly smooth, we know that is uniformly convex. In view of Lemma 2.7, we see that, for any ,

(3.18)

It follows that

(3.19)

Note that

(3.20)

It follows from (3.16) and (3.17) that

(3.21)

By (3.19), (3.21), and , we have

(3.22)

It follows from the property of that

(3.23)

Since as and is demicontinuous, we obtain that . Note that

(3.24)

This implies that

(3.25)

Since enjoys the Kadec-Klee property, we see that

(3.26)

Note that

(3.27)

From (3.23) and (3.26), we arrive at

(3.28)

Note that is demicontinuous. It follows that . On the other hand, since

(3.29)

by (3.28) we conclude that as . Since enjoys the Kadec-Klee property, we obtain that

(3.30)

By repeating (3.18)–(3.30), we also can get

(3.31)

(3.32)

(3.33)

Since each is closed, by (3.30) and (3.31) we conclude that , that is, , . On the other hand, Lemma 2.4, Remark 2.5, and Lemma 2.8 show that and are closed. So, by (3.32) and (3.33) we have and . Now, it follows from Lemma 2.4 and Remark 2.5 that () and (). Hence, () and (). Therefore, .

Finally, we prove that . From , by Lemma 2.1, we see that

(3.34)

Since for each , we have

(3.35)

Letting in (3.35), we see that

(3.36)

In view of Lemma 2.1, we can obtain that . This completes the proof.

Remark 3.2.

Obviously, the proof process of is simple since we replace the condition (A3) with (A3') which is such that and are closed. In fact, although the condition (A3') is stronger than (A3), it is not easier to verify the condition (A3) than verify the condition (A3'). Hence, from this point, the condition (A3') is acceptable. On the other hand, the definition of is of some interest.

If for each , for each and for each , then Theorem 3.1 reduces to the following result.

Corollary 3.3.

Let be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space which has the Kadec-Klee property. Let be a closed quasi--nonexpansive mapping, a bifunction satisfying the conditions (A1), (A2), (A3'), and (A4) and a continuous monotone mapping such that . Let . Let be a sequence defined by the following manner:

(3.37)

where and are defined by (1.11) and (1.12) with and , are the real numbers in satisfying . Then the sequence converges strongly to , where is the generalized projection from onto .

Corollary 3.4.

Let be a nonempty closed and convex subset of a Hilbert space . Let be a family of closed quasi-nonexpansive mappings, a finite family of bifunctions satisfying the conditions (A1)–(A4), and a finite family of continuous monotone mappings such that . Let . Define a sequence by the following manner:

(3.38)

where and are defined by (1.11) and (1.12) with ( is the identity mapping), are the real numbers in satisfying and for each , are the real numbers in satisfying . Then the sequence converges strongly to , where is the projection from onto .

Proof.

By the proof of Theorem 3.1, we have as ,

(3.39)

Since each is closed, we can conclude that , . Note that in a Hilbert space, a firmly-nonexpansive mapping is also nonexpansive. Hence, and are nonexpansive for each and . By demiclosed principle, we can conclude that and for each and . That is, . Then by the final part of proof of Theorem 3.1, we have . This completes the proof.

Let be a Hilbert space and a nonempty closed and convex subset of . A mapping is called a pseudocontraction if for all ,

(3.40)

or equivalently,

(3.41)

Let , where is a pseudocontraction. Then is a monotone mapping and . Moreover, . Indeed, it is easy to see that . Let . We have

(3.42)

for all . Take . Then we have . That is, . This shows that , which implies that . So, . Based this, we have following result.

Corollary 3.5.

Let be a nonempty closed and convex subset of a Hilbert space . Let be a family of closed quasi-nonexpansive mappings, a finite family of bifunctions satisfying the conditions (A1)–(A4), and a finite family of continuous pseudocontractions such that . Let . Define a sequence by the following manner:

(3.43)

where are defined by (1.11) with and is defined by

(3.44)

are the real numbers in satisfying and for each , are the real numbers in satisfying . Then the sequence converges strongly to , where is the projection from onto .

If , , and for each ,  , and , then Corollary 3.5 reduced the following result.

Corollary 3.6.

Let be a nonempty closed and convex subset of a Hilbert space . Let be a closed quasi-nonexpansive mapping,  a bifunction satisfying the conditions (A1)–(A4), and a continuous pseudocontraction such that . Let . Define a sequence by the following manner:

(3.45)

where is defined by (1.11) with and , is defined by (3.44) , and are the real numbers in satisfying . Then the sequence converges strongly to , where is the projection from onto .