#### Abstract

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative 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 Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. 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 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 Gáteaux differentiable norm. is said to have a uniformly Gáteaux 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 Fréchet differential if, for each , this limit is attained uniformly for . In addition, 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 . It is well-known that 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 Gáteaux differentiable norm then the duality mapping is norm-to-weak* uniformly continuous on bounded subsets of .

Let be a nonempty closed convex subset of a real Banach space . A mapping is called nonexpansive if

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.

*Definition 1. *Let be a mapping of into . Then is said to be(i)accretive if for each there exists such that
where is the normalized duality mapping;(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 be a real smooth Banach space. Let be a nonempty closed convex subset of and let be a nonlinear mapping. The so-called variational inequality problem (VIP) is the problem of finding such that which was considered by Aoyama et al. [1]. Note that VIP (10) is connected with the fixed point problem for nonlinear mapping (see e.g., [2]), the problem of finding a zero point of a nonlinear operator (see e.g., [3]), and so on. In particular, whenever a Hilbert space, the VIP (10) reduces to the classical VIP of finding such that

whose solution set is denoted by . Recently, in order to find a solution of VIP (10), Aoyama et al. [1] introduced Mann-type iterative scheme for an accretive operator as follows: where is a sunny nonexpansive retraction from onto . Then they proved a weak convergence theorem.

*Definition 2. *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 .

Lemma 3 (see [4]). *Let be a nonempty closed convex subset of a strictly convex Banach space. Let be a finite family of nonexpansive mappings of into itself with and let be real numbers such that for every and . Let be the -mapping generated by and . Then .*

From Lemma 3, it is easy to see that the -mapping is a nonexpansive mapping.

On the other hand, 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 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. [5] introduced and studied the multivalued variational inclusion (MVVI) of finding such that is a solution of the following:

If and , then the MVVI (15) reduces to the problem of finding such that is a solution of the following:

We denote by the set of such solutions for MVVI (16).

The authors [5] established an existence theorem for MVVI (15) in a smooth Banach space and then proved that the sequence generated by their iterative algorithm converges strongly to a solution of MVVI (16).

Theorem 4 (see [5, Theorem ]). *Let be a real smooth Banach space. Let , and 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 . Then, there exists such that, for and for all , converges strongly to , and, for any and , is a solution of the MVVI (16).*

Let be a nonempty closed convex subset of a real smooth Banach space and let be a sunny nonexpansive retraction from onto . Let be a contraction with coefficient . Motivated and inspired by the research going on this area, we introduce Mann-type viscosity approximation methods for finding solutions of the MVVI (16) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings. Here, the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative 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 Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of -strictly pseudocontractive mappings () and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. 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, [6–11].

#### 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(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 5 (see [12]). *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 6 (see [13]). *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* 2009 *is accretive with respect to the mapping* *.*

Then the mapping defined by is -strongly accretive.

Proposition 7 (see [14]). *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.*

Lemma 8 can be found in [15]. Lemma 9 is an immediate consequence of the subdifferential inequality of the function .

Lemma 8. *Let be a sequence of nonnegative real numbers satisfying
**
where , and satisfy the following conditions:*(i)* and ;*(ii)*;*(iii)*, for all , and .**Then .*

Lemma 9. *In a smooth Banach space , there holds the inequality
*

Lemma 10 (see [1]). *Let be a nonempty closed convex subset of a smooth Banach space . Let be a sunny nonexpansive retraction from onto and let be an accretive operator of into . Then, 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 11 (see [16]). *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 12 (see [17]). *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 13 (see [18]). *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 let be 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 14 (see [19]). *Let be a uniformly smooth Banach space or a reflexive and strictly convex Banach space with a uniformly Gáteaux differentiable norm. Let be a nonempty closed convex subset of , let be a nonexpansive mapping with , and let . Then the net defined by converges strongly to a point in . If one defines a mapping by , for all , then solves the VIP as follows:
*

Lemma 15 (see [20]). *Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose is nonempty. Let be a sequence of positive numbers with . Then a mapping on defined by for is defined well and nonexpansive, and holds.*

Lemma 16 (see [21]). *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 .*

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

In this section, we introduce Mann-type viscosity iterative algorithms in uniformly convex and -uniformly smooth Banach spaces and show strong convergence theorems. We will use the following useful lemma.

Lemma 17. *Let be a nonempty closed convex subset of a real -uniformly smooth Banach space . Let be an -inverse strongly accretive mapping. Then, one has
**
where . In particular, if , then is nonexpansive.*

Theorem 18. *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 be three multivalued mappings, let be a single-valued mapping, and let be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4. Consider that**(C6) is -inverse strongly accretive with .*

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 be a contraction with coefficient . Let be a countable family of nonexpansive mappings of into itself such that . Suppose that , and are the sequences in and satisfy the following conditions:(i);(ii) and ;(iii) for some ;(iv);(v);(vi).

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 strongly to , which solves the following VIP: and, for any and , is a solution of the MVVI (16).

*Proof. *First of all, by Lemma 17 we know that is a nonexpansive mapping, where for each . Hence, from the nonexpansivity of , it follows that is a nonexpansive mapping for each . Since is the -mapping generated by and , by Lemma 3, we deduce that . Utilizing Lemma 10, and the definition of , we get for each . Thus, we have

Now, let us show that for any , , there exists a point such that is a solution of the MVVI (15), for any and . Indeed, following the argument idea in the proof of Chidume et al. [5, Theorem 3.1], we put for all . Then by Proposition 6, is -strongly accretive. Since and are -uniformly continuous and is continuous, is continuous and hence lower semicontinuous. Thus, by Proposition 7, 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 [22, page 184], for any , we have that is closed and bounded. Therefore, by Morales [23], is surjective. Hence, for any and , there exists such that , where and . In addition, in terms of Proposition 7, we know that is a single-valued mapping. Assume that is -inverse strongly accretive with . Then by Lemma 17, we conclude that the mapping is 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 (35) it follows that
and hence

which immediately yields

Since is -strongly accretive with respect to and is accretive, we deduce from (41) that

Again from (35), we have that

Also, from Proposition 7, is a single-valued mapping; that is, for any and , we have and . On the other hand, it follows from Nadler [24] that, for and , there exist and such that
respectively. Therefore, from (42) and (36), we have

So, we get , a contradiction. Therefore, is bounded.

Let us show that and .

Indeed, we define by for all . Then, is a nonexpansive mapping and the iterative scheme (35) can be rewritten as follows:

Taking into account condition (iv), we may assume that for some . From (47), we can rewrite by
where . Now, we have
where for some . By simple calculation, we have

So, from (49), we get

Also, for convenience, we write

By simple calculation, we get

From (51) and (53), we deduce that
and hence