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Journal of Applied Mathematics

Volume 2012 (2012), Article ID 865810, 20 pages

http://dx.doi.org/10.1155/2012/865810

## On Variational Inclusion and Common Fixed Point Problems in *q*-Uniformly Smooth Banach Spaces

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 9 June 2012; Accepted 18 August 2012

Academic Editor: Alicia Cordero

Copyright © 2012 Yanlai Song et al. 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.

#### Abstract

We introduce a general iterative algorithm for finding a common element of the common fixed-point set of an infinite family of -strict pseudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in a *q*-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as the improvement, supplementation, development, and extension of the corresponding results in some references to a great extent.

#### 1. Introduction

Throughout this paper, we denote by and a real Banach space and the dual space of , respectively. Let be a subset of and a mapping on . We use to denote the set of fixed points of . Let be a real number. The (generalized) duality mapping is defined by for all , where denotes the generalized duality pairing between and . In particular, is called the normalized duality mapping and for . If is a Hilbert space, then , where is the identity mapping. It is well known that if is smooth, then is single-valued, which is denoted by .

The norm of a Banach space is said to be Gâteaux differentiable if the limit exists for all , on the unit sphere . If, for each , the limit (1.2) is uniformly attained for , then the norm of is said to be uniformly Gâteaux differentiable. The norm of is said to be Fréchet differentiable if, for each , the limit (1.2) is attained uniformly for .

Let be the modulus of smoothness of defined by

A Banach space is said to be uniformly smooth if as . Let . A Banach space is said to be -uniformly smooth, if there exists a fixed constant such that . It is well known that is uniformly smooth if and only if the norm of is uniformly Fréchet differentiable. If is -uniformly smooth, then and is uniformly smooth, and hence the norm of is uniformly Fréchet differentiable; in particular, the norm of is Fréchet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces are , where . More precisely, is -uniformly smooth for every .

A Banach space is said to be uniformly convex if, for any , there exists such that, for any , , implies . It is known that a uniformly convex Banach space is reflexive and strictly convex.

Recall that if and are nonempty subsets of a Banach space such that is nonempty closed convex and , then a mapping is sunny (see [1]) provided that for all and , whenever . A mapping is called a retraction if for all . Furthermore, is a sunny nonexpansive retraction from onto if is retraction from onto which is also sunny and nonexpansive. A subset of is called a sunny nonexpansive retraction of if there exists a sunny nonexpansive retraction from onto . The following proposition concerns the sunny nonexpansive retraction.

Proposition 1.1 (see [1]). *Let be a closed convex subset of a smooth Banach space . Let be a nonempty subset of . Let be a retraction and let be the normalized duality mapping on . Then the following are equivalent:*(a) is sunny and nonexpansive,(b), for all , ,(c), for all , .

Among nonlinear mappings, the classes of nonexpansive mappings and strict pseudocontractions are two kinds of the most important nonlinear mappings. The studies on them have a very long history (see, e.g., [1–29] and the references therein). Recall that a mapping is said to be nonexpansive, if

A mapping is said to be -strict pseudocontractive in the terminology of Browder and Petryshyn (see [2–4]), if there exists a constant such that for every , and for some . It is clear that (1.6) is equivalent to the following:

A mapping is said to be -Lipschitz if for all there exists a constant such that In particular, if , then is called contractive and if , then reduces to a nonexpansive mapping.

A mapping is said to be accretive if for all , there exists such that

For some , is said to be -strongly accretive if for all , , there exists such that

For some is said to be -inverse strongly accretive if for all , there exists such that

A set-valued mapping is said to be accretive if for any , , there exists , such that for all and

A set-valued mapping is said to be -accretive if is accretive and for every (equivalently, for some) , where is the identity mapping.

Let be -accretive. The mapping defined by is called the resolvent operator associated with , where is any positive number and is the identity mapping. It is well known that is single valued and nonexpansive (see [5]).

In order to find the common element of the solutions set of a variational inclusion and the set of fixed points of a nonexpansive mapping , Zhang et al. [6] introduced the following new iterative scheme in a Hilbert space . Starting with an arbitrary point , define sequences by where is an -cocoercive mapping, is a maximal monotone mapping, is a nonexpansive mapping, and is a sequence in . Under mild conditions, they obtained a strong convergence theorem.

Let be a nonempty closed convex subset of a real reflexive, strictly convex, and -uniformly smooth Banach space . In this paper, we consider the general system of finding such that where , and are nonlinear mappings.

In the case where , a uniformly convex and -uniformly smooth Banach space, Qin et al. [8] introduced the following scheme for finding a common element of the solution set of the variational inclusions and the fixed-point set of a -strict pseudocontraction. Starting with an arbitrary point , define sequences by where are two inverse strongly accretive operators, , are two maximal monotone mappings, is a -strict pseudocontraction, and is defined as , for all . Then they proved a strong convergence theorem under mild conditions.

In this paper, motivated by Zhang et al. [6], Qin et al. [8], Yao et al. [9], Hao [10], Yao and Yao [11], and Takahashi and Toyoda [12], we consider a relaxed extragradient-type method for finding a common element of the solution set of a general system of variational inclusions for inverse strongly accretive mappings and the common fixed-point set of an infinite family of -strict pseudocontractions. Furthermore, we obtain strong convergence theorems under mild conditions. The results presented by us improve and extend the corresponding results announced by many others.

#### 2. Preliminaries

In order to prove our main results, we need the following lemmas.

Lemma 2.1 (see [16]). *Let be a closed convex subset of a strictly convex Banach space . Let and be two nonexpansive mappings from into itself with . Define a mapping by
**
where is a constant in . Then is nonexpansive and .*

Lemma 2.2 (see [30]). *Let be a sequence of nonnegative numbers satisfying the property:
**
where , , and satisfy the restrictions: *(i)*, , *(ii)*, *(iii)*.**
Then, .*

Lemma 2.3 (see [31, page 63]). *Let . Then the following inequality holds:
**
for arbitrary positive real numbers , .*

Lemma 2.4 (see [17]). *Let be a real -uniformly smooth Banach space, then there exists a constant such that
**
In particular, if is a real 2-uniformly smooth Banach space, then there exists a best smooth constant such that
*

Lemma 2.5 (see [20]). *Let be a nonempty convex subset of a real -uniformly smooth Banach space and let be a -strict pseudocontraction. For , one defines . Then, as , , is nonexpansive such that .*

Lemma 2.6 (see [21]). *Let be a nonempty, closed, and convex subset of a real -uniformly smooth Banach space which admits weakly sequentially continuous generalized duality mapping from into . Let be a nonexpansive mapping. Then, for all , if and , then .*

Lemma 2.7 (see [21]). *Let be a nonempty, closed, and convex subset of a real -uniformly smooth Banach space . Let be a -Lipschitzian and -strongly accretive operator with constants . Let and . Then for each , the mapping defined by is a contraction with a constant .*

Lemma 2.8 (see [21]). *Let be a nonempty, closed and convex subset of a real -uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let be a -Lipschitzian and -strongly accretive operator with constants , a -Lipschitzian mapping with constant , and a nonexpansive mapping such that . Let and , where . Then defined by
**
Has the following properties:*(i)* is bounded for each ,*(ii)*,*(iii)* defines a continuous curve from into .*

Lemma 2.9. *Let be a closed convex subset of a smooth Banach space . Let be a nonempty subset of . Let be a retraction and let , be the normalized duality mapping and generalized duality mapping on , respectively. Then the following are equivalent:*(a)* is sunny and nonexpansive,*(b)*, for all , ,*(c)*, for all , ,*(d)*, for all , . *

* Proof. *From Proposition 1.1, we have . We need only to prove .

Indeed, if , then , for all , (by the fact that .

If , then , for all , . This completes the proof.

Lemma 2.10. *Let be a nonempty, closed, and convex subset of a -uniformly smooth Banach space which admits a weakly sequentially continuous generalized duality mapping from into . Let be a sunny nonexpansive retraction from onto . Let be a -Lipschitzian and -strongly accretive operator with constants , a -Lipschitzian with constant , and a nonexpansive mapping such that . Let and , where . For each , let be defined by (2.6), then converges strongly to as , which is the unique solution of the following variational inequality:
*

* Proof. *We first show the uniqueness of a solution of the variational inequality (2.7). Suppose both and are solutions of (2.7). It follows that
Adding up (2.8), we have
On the other hand, we have that
It is a contradiction. Therefore, and the uniqueness is proved. Below we use to denote the unique solution of (2.7).

Next, we prove that as .

Since is reflexive and is bounded due to Lemma 2.8 (i), there exists a subsequence of and some point such that . By Lemma 2.8(ii), we have . Taken together with Lemma 2.6, we can get that . Setting , where , then we can rewrite (2.6) as .

We claim .

From Lemma 2.9, we have
It follows from (2.11) and Lemma 2.7 that
It follows that
Therefore, we get
Using that the duality map is weakly sequentially continuous from to and noticing (2.14), we get that

We prove that solves the variational inequality (2.7). Since
we derive that
For all , note that
It follows from Lemma 2.9 and (2.18) that
where .

Now replacing in (2.19) with and letting , from (2.15) and Lemma 2.8 (ii), we obtain , that is, is a solution of (2.7). Hence by uniqueness. Therefore, as . And consequently, as .

Lemma 2.11. *Let be a nonempty closed convex subset of a real -uniformly smooth Banach space . Let the mapping be a -inverse-strongly accretive operator. Then the following inequality holds:
**
In particular, if , then is nonexpansive. *

* Proof. * Indeed, for all , it follows from Lemma 2.4 that
It is clear that if , then is nonexpansive. This completes the proof.

Lemma 2.12. *Let be a nonempty closed convex subset of a real -uniformly smooth Banach space . Suppose are two -accretive mappings and , are two arbitrary positive constants. Let be -inverse strongly accretive and -inverse strongly accretive, respectively. Let be a mapping defined by
**
If and , then is nonexpansive. *

* Proof. *For all , , by Lemma 2.11, we have
which implies that is nonexpansive. This completes the proof.

Lemma 2.13. *Let be a nonempty closed convex subset of a real -uniformly smooth Banach space . Suppose are two inverse strongly accretive operators, , are two -accretive mappings, and , are two arbitrary positive constants. Then is a solution of general system (1.15) if and only if , where is defined by Lemma 2.12.*

* Proof. *Note that
This completes the proof.

Lemma 2.14 (see [18]). *Let be a -uniformly smooth Banach space and a nonempty convex subset of . Assume for each , is a -strict pseudocontraction with . Assume and . Let be a positive sequence such that , then is a -strict pseudocontraction and .*

*Remark 2.15. *Under the assumptions of Lemma 2.14, if for each the mapping is replaced by , respectively, where is a nonempty closed convex subset of , then noticing the fact
by Lemma 2.14, we deduce that is a -strict pseudocontraction with and .

#### 3. Main Results

Theorem 3.1. *Let be a nonempty closed convex subset of a strictly convex, and uniformly smooth Banach space which admits a weakly sequentially continuous generalized duality mapping . Let be a sunny nonexpansive retraction from onto . Assume the mappings , are -inverse strongly accretive and -inverse strongly accretive, respectively. Let , two -accretive mappings and , be two arbitrary positive constants. Suppose is -Lipschitz and -strongly accretive with constants being -Lipschitz with constant . Let be an infinite family of -strict pseudocontractions with and . Let , , , , , where and . Assume and . Define a mapping , for all . For arbitrarily given and , let be the sequence generated iteratively by
**
Assume that , , and are three sequences in satisfying the following conditions: *(i)*,
*(ii)*,
*(iii)*. ** Suppose in addition that . Then converges strongly to some point , which is the unique solution of the following variational inequality:
*

* Proof. *We divide the proof into several steps. *Step 1*. First, we show that sequences are bounded. From and , there exist some such that . We may assume, without loss of generality, that . From Lemma 2.7, we deduce that
Taking , it follows from Lemma 2.13 that
Putting , then we can deduce that . By Lemma 2.11, we obtain
It follows from (3.5) that
In view of Remark 2.15, let be the mapping defined by for all , then we can deduce that is a -strict pseudocontraction and . By virtue of Lemma 2.5 and , where , we can get that is nonexpansive and . Putting , it follows that
It follows from (3.7) that
Hence, is bounded, so are , , , and .*Step 2*. In this part, we will claim that , as .

We observe that
It follows from (3.9) that
Again from (3.1), we have
It follows from (3.10) that
Substituting (3.12) into (3.11), we have
where . From (i), (ii), (iii), (3.13), and Lemma 2.2, we deduce that
We observe that
which implies that
Noticing conditions (i) and (ii) and (3.14), we have
Let
In view of Lemma 2.1, we see that is nonexpansive such that
Noticing that
one has
In view of (3.17), (iii) and (3.21), we deduce that
We define , then it follows from Lemma 2.10 that converges strongly to some point , which is the unique solution of the variational inequality (3.2).*Step 3.* We show that
where is the solution of the variational inequality of (3.2). To show this, we take a subsequence of such that
Without loss of generality, we may further assume that for some point due to reflexivity of the Banach space and boundness of , it follows from (3.22) and Lemma 2.6 that . Since the Banach space has a weakly sequentially continuous generalized duality mapping , we obtain that
*Step 4.* We prove that . Setting , for all . It follows from (3.1) that . In view of Lemmas 2.3, 2.7, and 2.9, we have
which implies
Put and . Apply Lemma 2.2 to (3.27) to obtain as . This completes the proof.

*Remark 3.2. *Compared with the known results in the literature, our results are very different from those in the following aspects. (i)The results in this paper improve and extend corresponding results in [6–13]. Especially, our result extends their results from -uniformly smooth Banach space or Hilbert space to more general -uniformly smooth Banach space.(ii)Our Theorem 3.1 extends one nonexpansive mapping in [6, Theorem 2.1], one -strict pseudocontraction in [8, Theorem 3.1], and an infinite family of nonexpansive mappings in [10, Theorem 3.1] to an infinite family of -strict pseudocontractions. And our Theorem 3.1 gets a common element of the common fixed-point set of an infinite family of -strict pseudocontractions and the solution set of the general system of variational inclusions for two inverse strongly accretive mappings in a -uniformly smooth Banach space.(iii)We by replace the which is a fixed element in iterative scheme (1.16), where is a -Lipschitzian. And we also add a Lipschitz and strong accretive operator in our scheme (3.1). In particular, whenever , , , and , our scheme (3.1) reduces to (1.16). (iv)It is worth noting that the Banach space does not have to be uniformly convex in our Theorem 3.1. However, it is very necessary in Theorem 3.1 of Qin et al. [8] and many other literature.

Corollary 3.3. *Let be a nonempty closed convex subset of a strictly convex, and -reflexive which admits a weakly sequentially continuous normalized duality mapping . Let be a sunny nonexpansive retraction from onto . Assume the mappings , are -inverse strongly accretive and -inverse strongly accretive, respectively. Let , be two -accretive operators and , two arbitrary positive constants. Suppose is a -Lipschitzian and -strongly accretive operator with constants , , being a -Lipschitzian with constant . Let , , and , where . Let be a nonexpansive with . For arbitrarily given *