## Preserver Problems on Function Spaces, Operator Algebras, and Related Topics

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# Hybrid and Relaxed Mann Iterations for General Systems of Variational Inequalities and Nonexpansive Mappings

**Academic Editor:**Jen-Chih Yao

#### Abstract

We introduce hybrid and relaxed Mann iteration methods for a general system of variational inequalities with solutions being also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in real smooth and uniformly convex Banach spaces. Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

#### 1. Introduction

Let be a real Banach space whose dual space is denoted by . The normalized duality mapping is defined by

where denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that is nonempty for each . Let be a nonempty closed convex subset of . A mapping is called nonexpansive if for every . The set of fixed points of is denoted by . We use the notation to indicate the weak convergence and the one to indicate the strong convergence. A mapping is said to be(i) accretive if for each there exists such that (ii) -strongly accretive if for each there exists such that for some ;(iii) -inverse strongly accretive if, for each , there exists such that for some ;(iv) -strictly pseudocontractive [1] (see also [2]) if for each there exists such that for some .

It is worth emphasizing that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for example, [3–5]. Let denote the unite sphere of . A Banach space is said to be uniformly convex if, for each , there exists such that, for all ,

It is known that a uniformly convex Banach space is reflexive and strict convex. A Banach space is said to be smooth if the limit

exists for all ; in this case, is also said to have a Gateaux differentiable norm. is said to have a uniformly, Gateaux differentiable norm if, for each , the limit is attained uniformly for . Moreover, it is said to be uniformly smooth if this limit is attained uniformly for . The norm of is said to be the Frechet differential if for each , this limit is attained uniformly for . In the meantime, we define a function called the modulus of smoothness of as follows:

It is known that is uniformly smooth if and only if . Let be a fixed real number with . Then, a Banach space is said to be -uniformly smooth if there exists a constant such that for all . As pointed out in [6], no Banach space is -uniformly smooth for . In addition, it is also known that is single valued if and only if is smooth, whereas if is uniformly smooth, then the mapping is norm-to-norm uniformly continuous on bounded subsets of . If has a uniformly Gateaux differentiable norm, then the duality mapping is norm-to-weak* uniformly continuous on bounded subsets of .

Recently, Yao et al. [7] combined the viscosity approximation method and Mann iteration method and gave the following hybrid viscosity approximation method.

Let be a nonempty closed convex subset of a real uniformly smooth Banach space , a nonexpansive mapping with , and a contraction with coefficient . For an arbitrary , define in the following way:

where and are two sequences in .

They proved under certain control conditions on the sequences and that converges strongly to a fixed point of . Subsequently, under the following control conditions on and :(i), for all for some integer ,(ii),(iii),(iv).

Ceng and Yao [8] proved that

where solves the variational inequality problem (VIP):

Such a result includes [7, Theorem 1] as a special case.

Let be a nonempty closed convex subset of a real Banach space and with a contractive coefficient , where is the set of all contractive self-mappings on . Let be a sequence of nonexpansive self-mappings on and a sequence of nonnegative numbers in . For any , define a self-mapping on as follows:

Such a mapping is called the -mapping generated by , and ; see [9].

In 2008, Ceng and Yao [10] introduced and analyzed the following relaxed viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in a strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm.

Theorem 1 (see [10]). *Let be a strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm, a nonempty closed convex subset of , a sequence of nonexpansive self-mappings on such that the common fixed point set , and with a contractive coefficient . For any given , let be the iterative sequence defined by
**
where and are two sequences in with , is a sequence in , and is the -mapping generated by . Assume that*(i)*, and ;*(ii)* and .**Then, there hold the following:*(i)*;*(ii)*the sequence converges strongly to some which is the unique solution of the variational inequality problem (VIP)
* *provided and for some fixed .*

On the other hand, Cai and Bu [11] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space , which involves finding such that

where is a nonempty, closed, and convex subset of , are two nonlinear mappings, and and are two positive constants. Here, the set of solutions of GSVI (13) is denoted by . In particular, if , a real Hilbert space, then GSVI (13) reduces to the following GSVI of finding such that

in which and are two positive constants. The set of solutions of problem (14) is still denoted by . In particular, if , then problem (14) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [12]. Further, if additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding such that

The solution set of the VIP (15) is denoted by . Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [13]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous.

Recently, Ceng et al. [14] transformed problem (14) into a fixed point problem in the following way.

Lemma 2 (see [14]). *For given , is a solution of problem (14) if and only if is a fixed point of the mapping defined by
**
where and is the projection of onto .*

In particular, if the mapping is -inverse strongly monotone for , then the mapping is nonexpansive provided for .

In 1976, Korpelevič [15] proposed an iterative algorithm for solving the VIP (15) in Euclidean space :

with a given number, which is known as the extragradient method (see also [16]). The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [3, 11, 13, 17–33] and references therein, to name but a few.

In particular, whenever is still a real smooth Banach space, and , then GSVI (13) reduces to the variational inequality problem (VIP) of finding such that

which was considered by Aoyama et al. [34]. Note that VIP (18) is connected with the fixed point problem for nonlinear mapping (see, e.g., [35]), the problem of finding a zero point of a nonlinear operator (see, e.g., [36]), and so on. It is clear that VIP (18) extends VIP (15) from Hilbert spaces to Banach spaces.

In order to find a solution of VIP (18), Aoyama et al. [34] introduced the following Mann-type iterative scheme for an accretive operator :

where is a sunny nonexpansive retraction from onto . Then, they proved a weak convergence theorem. For the related work, see [37] and the references therein.

Let be a nonempty convex subset of a real Banach space . Let be a finite family of nonexpansive mappings of into itself and let be real numbers such that for every . Define a mapping as follows:

Such a mapping is called the -mapping generated by and .

Very recently, Kangtunyakarn [38] introduced and analyzed an iterative algorithm by the modification of Mann’s iteration process for finding a common element of the set of solutions of a finite family of variational inequalities and the set of fixed points of an -strictly pseudocontractive mapping and a nonexpansive mapping in uniformly convex and -uniformly smooth Banach spaces.

Theorem 3 (see [38]). *Let be a nonempty closed convex subset of a uniformly convex and -uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let be an -inverse-strongly accretive mapping for each . Define the mapping by for , where and is the -uniformly smooth constant of . Let be the -mapping generated by and , where , for all , and . Let a contraction with coefficient . Let be an -strictly pseudocontractive mapping and be a nonexpansive mapping such that . For arbitrarily given , let be the sequence generated by
**
where . Suppose that , , , and are the sequences in , and satisfy the following conditions:*(i)* and ;*(ii)* for some ;*(iii)*;*(iv)*.**
Then, converges strongly to , which solves the following VIP:
*

Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (13) which contains VIP (18) as a special case. Very recently, Cai and Bu [11] constructed an iterative algorithm for solving GSVI (13) and a common fixed point problem of a countable family of nonexpansive mappings in a uniformly convex and -uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a -uniformly smooth Banach space .

Lemma 4 (see [39]). *Let be a -uniformly smooth Banach space. Then,
**
where is the -uniformly smooth constant of and is the normalized duality mapping from into .*

Define the mapping as follows:

The fixed point set of is denoted by . Then, their strong convergence theorem on the proposed method is stated as follows.

Theorem 5 (see [11]). *Let be a nonempty closed convex subset of a uniformly convex and -uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse-strongly accretive with for . Let be a contraction of into itself with coefficient . Let be a countable family of nonexpansive mappings of into itself such that , where is the fixed point set of the mapping defined by (24). For arbitrarily given , let be the sequence generated by
*

Suppose that and are two sequences in satisfying the following conditions:(i) and ;(ii).

Assume that for any bounded subset of and let be a mapping of into defined by for all and suppose that . Then, converges strongly to , which solves the following VIP:

It is easy to see that the iterative scheme in Theorem 5 is essentially equivalent to the following two-step iterative scheme:

For the convenience of implementing the argument techniques in [14], the authors of [11] have used the following inequality in a real smooth and uniform convex Banach space .

Proposition 6 (see [40]). *Let be a real smooth and uniform convex Banach space and let . Then, there exists a strictly increasing, continuous, and convex function , such that
**
where .*

Let be a nonempty closed convex subset of a real smooth Banach space . Let be a sunny nonexpansive retraction from onto and a contraction with coefficient . Motivated and inspired by the research going on this area, we consider and introduce hybrid and relaxed Mann iteration methods for finding solutions of the GSVI (13) which are also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in . Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and -uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature; see, for example, [8, 10, 11, 14, 33, 38].

#### 2. Preliminaries

We list some lemmas that will be used in the sequel.

Lemma 7 (see [41]). *Let be a sequence of nonnegative real numbers satisfying
**
where , , and satisfy the following conditions:*(i)* and ;*(ii)*;*(iii)*, for all , and .**
Then, .*

The following lemma is an immediate consequence of the subdifferential inequality of the function .

Lemma 8 (see [42]). *Let be a real Banach space . Then, for all *(i)* for all ;*(ii)* for all .*

Let be a subset of and let be a mapping of into . Then, is said to be sunny if

whenever for and . A mapping of into itself is called a retraction if . If a mapping of into itself is a retraction, then for every where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto . The following lemma concerns the sunny nonexpansive retraction.

Lemma 9 (see [43]). *Let be a nonempty closed convex subset of a real smooth Banach space . Let be a nonempty subset of . Let be a retraction of onto . Then, the following are equivalent:*(i)* is sunny and nonexpansive;*(ii)* for all ;*(iii)*, for all , .*

It is well known that if a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from onto ; that is, . If is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space and if is a nonexpansive mapping with the fixed point set , then the set is a sunny nonexpansive retract of .

Lemma 10. *Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto and let be nonlinear mappings. For given , is a solution of GSVI (13) if and only if , where .*

*Proof. *We can rewrite GSVI (13) as

which is obviously equivalent to

because of Lemma 9. This completes the proof.

In terms of Lemma 10, we observe that

which implies that is a fixed point of the mapping . Throughout this paper, the set of fixed points of the mapping is denoted by .

Lemma 11 (see [44]). *Let be a uniformly convex Banach space and , . Then, there exists a continuous, strictly increasing, and convex function , such that
**
for all , and all with .*

Lemma 12 (see [45]). *Let be a nonempty closed convex subset of a Banach space . Let be a sequence of mappings of into itself. Suppose that . Then for each , converges strongly to some point of . Moreover, let be a mapping of into itself defined by for all . Then .*

Let be a nonempty closed convex subset of a Banach space and a nonexpansive mapping with . As previous, let be the set of all contractions on . For and , let be the unique fixed point of the contraction on ; that is,

Lemma 13 (see [35, 46]). *Let be a uniformly smooth Banach space, or a reflexive and strictly convex Banach space with a uniformly Gateaux differentiable norm. Let be a nonempty closed convex subset of , a nonexpansive mapping with , and . Then, the net defined by converges strongly to a point in . If we define a mapping by , for all , then solves the VIP:
*

Lemma 14 (see [47]). *Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose that is nonempty. Let be a sequence of positive numbers with . Then, a mapping on defined by for is defined well; nonexpansive and holds.*

Lemma 15 (see [39]). *Given a number , A real Banach space is uniformly convex if and only if there exists a continuous strictly increasing function , , such that
**
for all and such that and .*

Lemma 16 (see [48, Lemma 3.2]). *Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive self-mappings on such that and let be a sequence of positive numbers in for some . Then, for every and , the limit exists.*

Using Lemma 16, one can define a mapping as follows:

for every . Such a is called the -mapping generated by the sequences and . Throughout this paper, we always assume that is a sequence of positive numbers in for some .

Lemma 17 (see [48]). *Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive self-mappings on such that and let be a sequence of positive numbers in for some . Then, .**Let be a continuous linear functional on and . One writes instead of . is called a Banach limit if satisfies and for all . If is a Banach limit, then, there hold the following:*(i)*for all , implies ;*(ii)* for any fixed positive integer ;*(iii)* for all .*

Lemma 18 (see [49]). *Let be a real number and a sequence satisfy the condition for all Banach limit . If , then .*

In particular, if in Lemma 18, then we immediately obtain the following corollary.

Corollary 19 (see [50]). *Let be a real number and a sequence satisfy the condition for all Banach limit . If , then, .*

Lemma 20 (see [51]). *Let and be bounded sequences in a Banach space and let be a sequence of nonnegative numbers in with . Suppose that for all integers and . Then, .*

Lemma 21 (see [34]). *Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto and an accretive mapping. Then for all ,
*

Lemma 22 (see [11]). *Let be a nonempty closed convex subset of a real -uniformly smooth Banach space . Let the mapping be -inverse-strongly accretive. Then, one has
**
for where . In particular, if , then is nonexpansive for .*

Lemma 23 (see [11]). *Let be a nonempty closed convex subset of a real -uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let the mapping be -inverse-strongly accretive for . Let be the mapping defined by
**
If for , then is nonexpansive.*

#### 3. Hybrid Mann Iterations and Their Convergence Criteria

In this section, we introduce our hybrid Mann iteration algorithms in real smooth and uniformly convex Banach spaces and present their convergence criteria.

Theorem 24. *Let be a nonempty closed convex subset of a uniformly convex and -uniformly smooth Banach space . Let be a sunny nonexpansive retraction from onto . Let be a sequence of positive numbers in for some and an -inverse strongly accretive mapping for each . Define a mapping by for all and , where , is the -uniformly smooth constant of . Let be the -mapping generated by and . Let the mapping be -inverse strongly accretive for . Let be a contraction with coefficient . Let be a countable family of nonexpansive mappings of into itself such that , where is the fixed point set of the mapping with for . For arbitrarily given , let be the sequence generated by
**
where , , , and are the sequences in such that for all . Suppose that the following conditions hold:*(i)* and , for all for some integer ;*(ii)* and ;*(iii)*;*(iv)*.**Assume that for any bounded subset of and let be a mapping of into itself defined by for all and suppose that . Then, there hold the following:*(I)*;*(II)* provided for some fixed , where solves the following VIP:
*

*Proof. *First of all, since for , it is easy to see that is a nonexpansive mapping for each . Since is the -mapping generated by and , by Lemma 16 we know that, for each and , the limit exists. Moreover, one can define a mapping as follows:

for every . That is, such a is the -mapping generated by the sequences and . According to Lemma 17, we know that . From Lemma 15 and the definition of , we have for each . Hence, we have

Next, let us show that the sequence is bounded. Indeed, take a fixed arbitrarily. Then, we get , , and for all . By Lemma 23 we know that is nonexpansive. Then, from (42), we have

and hence

By induction, we obtain

Thus, is bounded, and so are the sequences , and .

Let us show that

As a matter of fact, put , for all . Then, it follows from (i) and (iv) that

and hence

Define

Observe that

and hence