#### Abstract

We introduce Mann-type extragradient methods for a general system of variational inequalities with solutions of a multivalued variational inclusion and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. 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 as follows:(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 if for each there exists such that for some .

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 strictly convex. A Banach space is said to be smooth if the limit exists for all ; in this case, is also said to have a Gáteaux differentiable norm. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for . The norm of is said to be the Fréchet 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 [1], 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 is norm-to-norm uniformly continuous on bounded subsets of .

Very recently, Cai and Bu [2] 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 (9) is denoted by . In particular, if , a real Hilbert space, then GSVI (9) reduces to the following GSVI of finding such that where and are two positive constants. The set of solutions of problem (10) is still denoted by .

Recently, Ceng et al. [3] transformed problem (10) into a fixed point problem in the following way.

Lemma 1 (see [3]). *For given is a solution of problem (10) if and only if is a fixed point of the mapping defined by
**
where and is the the projection of onto .*

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

Define the mapping as follows: The fixed point set of is denoted by .

Let be the family of all nonempty, closed and bounded subsets of a real smooth Banach space . Also, we denote by the Hausdorff metric on defined by Let and be two multivalued mappings, let be an -accretive mapping, let be a single-valued mapping, and let be a nonlinear mapping. Then for any given , , Chidume et al. [4] introduced and studied the multivalued variational inclusion (MVVI) of finding such that is a solution of the following:

If and , then the MVVI (14) reduces to the problem of finding such that is a solution of the following: We denote by the set of such solutions for MVVI (15).

The authors [4] first established an existence theorem for MVVI (14) in smooth Banach space and then proved that the sequence generated by their iterative algorithm converges strongly to a solution of MVVI (15).

Theorem 2 (see [4, Theorem 3.2]). *Let be a real smooth Banach space. Let and , let be three multivalued mappings, let be a single-valued mapping, and let be a single-valued continuous mapping satisfying the following conditions:*(C1)* is -accretive and -uniformly continuous;*(C2)* is -uniformly continuous;*(C3)* is -uniformly continuous;*(C4)*the mapping is -strongly accretive and --Lipschitz with respect to the mapping , where is a strictly increasing function with ;*(C5)*the mapping is accretive and --Lipschitz with respect to the mapping .** For arbitrary define the sequence iteratively by
**
where is defined by
**
for any , and some , where is a positive real sequence such that and . Then, there exists such that for , for all , converges strongly to ; and for any , , is a solution of the MVVI (15).*

Let be a nonempty closed convex subset of a real smooth Banach space and let be a sunny nonexpansive retraction from onto . Motivated and inspired by the research going on this area, we introduce Mann-type extragradient methods for finding solutions of the GSVI (9) which are also ones of the MVVI (15) and common fixed points of a countable family of nonexpansive mappings. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and -uniformly smooth Banach space, and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. 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, [2–10].

#### 2. Preliminaries

Let be a real Banach space with dual . We denote by the normalized duality mapping from to defined by where denotes the generalized duality pairing. Throughout this paper the single-valued normalized duality map is still denoted by . Unless otherwise stated, we assume that is a smooth Banach space with dual .

A multivalued mapping is said to be as follows:(i)accretive, if (ii)-accretive, if is accretive and , for all , where is the identity mapping;(iii)-inverse strongly accretive, if there exists a constant such that (iv)-strongly accretive, if there exists a strictly increasing continuous function with such that (v)-expansive, if

It is easy to see that if is -strongly accretive, then is -expansive.

A mapping is said to be -uniformly continuous, if for any given there exists a such that whenever then .

A mapping is -strongly accretive, with respect to , in the first argument if

A mapping is called lower semicontinuous if is open in whenever is open.

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

Proposition 3 (see [11]). *Let and be sequences of nonnegative numbers and a sequence satisfying the conditions that is bounded, , and , as . Let the recursive inequality
**
be given where is a strictly increasing function such that it is positive on and . Then , as .*

Proposition 4 (see [12]). *Let be a real smooth Banach space. Let and be two multivalued mappings, and let be a nonlinear mapping satisfying the following conditions:**(i)** the mapping ** is **-strongly accretive with respect to the mapping **;**(ii)** the mapping ** is accretive with respect to the mapping **.** Then the mapping defined by is -strongly accretive.*

Proposition 5 (see [13]). *Let be a real Banach space and let be a lower semicontinuous and -strongly accretive mapping; then for any is a one-point set; that is, is a single-valued mapping.*

Recall that a Banach space is said to satisfy Opial’s condition, if whenever is a sequence in which converges weakly to as , then

Lemma 6 (Demiclosedness principle; see [14, Lemma 2]). *Let be a nonempty closed convex subset of a reflexive Banach space that satisfies Opial’s condition and suppose that is nonexpansive. Then the mapping is demiclosed at zero, that is, and imply ; that is, .*

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

Lemma 7. *In a real smooth Banach space , there holds the inequality
**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 8 (see [15]). *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 . The following result is an easy consequence of Lemma 8.

Lemma 9. *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 (9) if and only if , where .*

In terms of Lemma 9, 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 10 (see [16]). *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 11 (see [17]). *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 .*

#### 3. Mann-Type Extragradient Algorithms in Uniformly Convex and 2-Uniformly Smooth Banach Spaces

In this section, we introduce Mann-type extragradient algorithms in uniformly convex and 2-uniformly smooth Banach spaces and show weak and strong convergence theorems. We will use some useful lemmas in the sequel.

Lemma 12 (see [2, Lemma 2.8]). *Let be a nonempty closed convex subset of a real -uniformly smooth Banach space . Let the mapping be -inverse-strongly accretive. Then, we have
**
for , where . In particular, if , then is nonexpansive for .*

Lemma 13 (see [2, Lemma 2.9]). *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.*

Theorem 14. *Let be a uniformly convex and -uniformly smooth Banach space satisfying Opial’s condition and let be a nonempty closed convex subset of such that . Let be a sunny nonexpansive retraction from onto . Let and and let be three multivalued mappings, let be a single-valued mapping, and let be a single-valued continuous mapping satisfying conditions in Theorem 2 and*(C6)* is -inverse strongly accretive with .** Let be -inverse strongly accretive for . Let be a countable family of nonexpansive mappings of into itself such that , where is the fixed point set of the mapping with for . Assume that , , and are sequences in such that*(i)*;*(ii)*;*(iii)*.** For arbitrary define the sequence iteratively by
**
where is defined by
**
for any , and some . Assume that for any bounded subset of and let be a mapping of into itself defined by for all and suppose that . Then converges weakly to some , and for any , is a solution of the MVVI (15).*

*Proof. *First of all, let us show that for any , , there exists a point such that is a solution of the MVVI (14), for any and . Indeed, following the argument idea in the proof of Chidume et al. [4, Theorem 3.1], we put for all . Then by Proposition 4, is -strongly accretive. Since and are -uniformly continuous and is continuous, is continuous and hence lower semicontinuous. Thus, by Proposition 5 is single-valued. Moreover, since is -strongly accretive and by assumption is -accretive, we have that is an -accretive and -strongly accretive mapping, and hence by Cioranescu [18, page 184] for any we have is closed and bounded. Therefore, by Morales [19], is surjective. Hence, for any and there exists such that , where and . In addition, in terms of Proposition 5 we know that is a single-valued mapping. Assume that is -inverse strongly accretive with . Then by Lemma 12, we conclude that the mapping is nonexpansive. Meantime, by Lemma 13 we know that is also nonexpansive.

Without loss of generality we may assume that and . Let and let be sufficiently large such that . Then such that for any and . Let . Then as and are -uniformly continuous on , for , and , , there exist such that for any , , and imply , and , respectively.

Let us show that for all . We show this by induction. First, by construction. Assume that . We show that . If possible we assume that , then . Further from (32) it follows that
and hence
which immediately yields
Since is -strongly accretive with respect to and is accretive, we deduce from (36) that
Again from (32) we have that
Also, from Proposition 5, is a single-valued mapping; that is, for any and we have and . On the other hand, it follows from Nadler [20] that, for and , there exist and such that
respectively. Therefore, from (37) and (33), we have that
So, we get , a contradiction. Therefore, is bounded.

Let us show that and .

Indeed, utilizing Lemma 10 and the nonexpansivity of the mapping , we obtain from (32) that for all
It is easy to see that the limit exists. Meantime, it can be readily seen from (41) that
which together with conditions (i) and (iii) and the existence of , implies that
Utilizing the properties of and , we get
Note that
So, we have
Also, observe that
Thus, from (44) and (46) it follows that

Let us show that .

Indeed, for simplicity, put , and . Then for all . From Lemma 12 we have
Substituting (49) for (50), we obtain
Utilizing [21, Proposition 1] and Lemma 10, from (32) and (51) we have
which hence implies that
Since for and , are bounded, we obtain from (46), (53), condition (ii) and the properties of that
Utilizing Proposition 3 and Lemma 8, we have
which implies that
In the same way, we derive
which implies that
Substituting (56) for (58), we get
By Lemma 7, we have from (52) and (59)
which hence leads to
From (46), (54), (61), condition (ii), and the boundedness of , , , and , we deduce that
Utilizing the properties of and , we deduce that
From (63) we get
which together with (54), leads to
Since
Utilizing the assumption on and Lemma 11, from (65) we get

Next, let us show that converges weakly to some .

Indeed, since is reflexive and is bounded, there exists a subsequence of such that . Then by Lemma 6, we obtain from (44), (65), and (67) that , , and . Thus, . In addition, if is another subsequence of such that , then by Lemma 6 we also deduce from (44), (65), and (67) that . Thus, the limits and exist. Now we claim that . Assume that . Then in terms of Opial’s condition, we get
which leads to a contradiction. So, we must have . Therefore, . This completes the proof.

Theorem 15. *Let be a uniformly convex and -uniformly smooth Banach space and let be a nonempty closed convex subset of such that . Let be a sunny nonexpansive retraction from onto . Let and and let be three multivalued mappings, let be a single-valued mapping, and let be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 2. Let be -inverse strongly accretive for . Let be a countable family of nonexpansive mappings of into itself such that , where is the fixed point set of the mapping with *