#### Abstract

We propose a new viscosity iterative scheme for finding fixed points of nonexpansive mappings in a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of the space has the fixed point property for nonexpansive mappings. Certain different control conditions for viscosity iterative scheme are given and strong convergence of viscosity iterative scheme to a solution of a ceratin variational inequality is established.

#### 1. Introduction

Let be a real Banach space and let be a nonempty closed convex subset of . Recall that a mapping is a * contraction* on if there exists a constant such that We use to denote the collection of mappings verifying the above inequality. That is, Let be a nonexpansive mapping (recall that a mapping is * nonexpansive* if and let denote the set of fixed points of ; that is,

We consider the iterative scheme: for a nonexpansive mapping, and , As a special case of (1.1), the following iterative scheme: where are arbitrary (but fixed), has been investigated by many authors; see, for example, Browder [1], Chang [2], Cho et al. [3], Halpern [4], Lions [5], Reich [6, 7], Shioji and Takahashi [8], Wittmann [9], and Xu [10]. The authors above showed that the sequence generated by (1.2) converges strongly to a point in the fixed-point set under appropriate conditions on in Hilbert spaces or certain Banach spaces.

The viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping in a Hilbert space was proposed by Moudafi [11] in 2000. In 2004, Xu [12] extended Theorem 2.2 of Moudafi [11] for the iterative scheme (1.1) to a Banach space setting by using the following conditions on the sequence : For the iterative scheme (1.1) with generalized contractive mappings instead of contractions, we refer to [13].

In 2005, Kim and Xu [14] provided a simpler modification of Mann iterative scheme in a uniformly smooth Banach space as follows: where is an arbitrary (but fixed) element, and and are two sequences in (0,1). They proved that the sequence generated by (1.4) converges to a fixed point of under the following control conditions:

(i)(ii)(iii)Recently, Yao et al. [15] considered the following modified Mann iterative scheme in a uniformly smooth Banach space as the viscosity approximation method: and proved strong convergence of the sequence generated by (1.5) under certain different control conditions on and . In particular, their results remove the condition imposed on .

Very recently, Qin et al. [16] proposed the composite Halpern type iterative scheme in a uniformly smooth Banach space as follows: and showed strong convergence of the sequence generated by (1.6) under the following control conditions:

(i)(ii)(iii)In this paper, under the framework of a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of has the fixed point property for nonexpansive mappings, we consider a new composite iterative scheme for a nonexpansive mapping as the viscosity approximation method: for and the initial guess , where , and are sequences in . First, we prove under certain control conditions on the sequences , and different from those of Qin et al. [16] that the sequence generated by (IS) converges strongly to a fixed point of , which is a solution of a certain variational inequality. Next we study the composite iterative scheme (IS) with the weakly contractive mapping instead of the contractions. The main results develop and complement the corresponding results of [2, 3, 8, 9, 11, 12, 15, 16]. In particular, if for all in (IS), then (IS) reduces a new viscosity iterative scheme for finding a fixed point of :

#### 2. Preliminaries and Lemmas

Let be a real Banach space with norm , and let be its dual. The value of at will be denoted by . When is a sequence in , then (resp., ) will denote strong (resp., weak) convergence of the sequence to .

The (* normalized*) * duality mapping* from into the family of nonempty (by Hahn-Banach theorem) weak-star compact subsets of its dual is defined by
for each [17].

The norm of is said to be * Gâteaux differentiable* (and is said to be * smooth*) if
exists for each in its unit sphere . The norm is said to be * uniformly Gâteaux differentiable* if for , the limit is attained uniformly for . The space is said to have a * uniformly Fréchet differentiable norm* (and is said to be * uniformly smooth*) if the limit in (2.2) is attained uniformly for . It is known that is smooth if and only if each duality mapping is single-valued. It is also well-known that if has a uniformly Gâteaux differentiable norm, is uniformly norm-to- continuous on each bounded subsets of [17].

Let be a nonempty closed convex subset of . is said to have the * fixed point property* for nonexpansive mappings if every nonexpansive mapping of a bounded closed convex subset of has a fixed point in . Let be a subset of . Then a mapping is said to be a * retraction* from onto if for all . A retraction is said to be * sunny* if for all and with . A subset of is said to be a * sunny nonexpansive retract* of if there exists a sunny nonexpansive retraction of onto . In a smooth Banach space , it is well-known [18, page 48] that is a sunny nonexpansive retraction from onto if and only if the following condition holds
We need the following lemmas for the proof of our main results. Lemma 2.1 was also given in Jung and Morales [19], Lemma 2.2 is Lemma 2 of Suzuki [20] and Lemma 2.3 is essentially Lemma 2 of Liu [21] (also see [10]).

Lemma 2.1. *Let ** be a real Banach space and let ** be the duality mapping. Then, for any given *, * one has**
for all .*

Lemma 2.2. *Let ** and ** be bounded sequences in a Banach space ** and let ** be a sequence in ** which satisfies the following condition:**
suppose that
**
Then *

Lemma 2.3. *Let ** be a sequence of nonnegative real numbers satisfying**
where , , and satisfy the following conditions: *(i)* and or, equivalently, *(ii)* or *(iii)*. **Then .*

Recall that a mapping is said to be *weakly contractive* if
where is a continuous and strictly increasing function such that is positive on and . As a special case, if for , where , then the weakly contractive mapping is a contraction with constant . Rhoades [22] obtained the following result for weakly contractive mapping.

Lemma 2.4 ([22, Theorem 2]). *Let ** be a complete metric space and let ** be a weakly contractive mapping on **. Then ** has a unique fixed point ** in *. *Moreover, for *, * converges strongly to **.*

The following Lemma was given in [23, 24].

Lemma 2.5. * Let and be two sequences of nonnegative real numbers and let be a sequence of positive numbers satisfying the conditions:*(i)

*;*(ii)

*.*

*Let the recursive inequality,*

*be given, where is a continuous and strict increasing function on with . Then .*

#### 3. Main Results

First, we study a strong convergence theorem for a viscosity iterative scheme for the nonexpansive mapping with the contraction.

For a nonexpansive mapping, and , defines a strict contraction mapping. Thus, by the Banach contraction mapping principle, there exists a unique fixed point satisfying

For simplicity we will write for provided no confusion occurs.

In 2006, the following result was given by Jung [25] (see also Xu [12] for the result in uniformly smooth Banach spaces).

Theorem 3 J (see [25]). *Let be a reflexive Banach space having a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of has the fixed point property for nonexpansive mappings. Let be a nonempty closed convex subset of and let be a nonexpansive mapping from into itself with Then defined by (R) converges strongly to a point in If we define by
**
then is the unique solution of the variational inequality
*

*Remark 3.1. *In Theorem J, if is a constant, then (VI) become
Hence by (2.3), reduces to the sunny nonexpansive retraction from to . Namely is a sunny nonexpansive retraction of .

Using Theorem J, we have the following result.

Theorem 3.2. *Let be a reflexive Banach space having a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of has the fixed point property for nonexpansive mappings. Let be a nonempty closed convex subset of and let be a nonexpansive mapping from into itself with .Let , and be sequences in which satisfy the conditions: *(C1)

*, ;*(C2)

*;*(C3)

*Let and the initial guess be chosen arbitrarily. Let be the sequence generated by*

*If , then converges strongly to , where is the unique solution of the variational inequality*

*Proof. *We note that by Theorem J, there exists the unique solution of the variational inequality
Namely, where is defined by (R). We will show that .

We proceed with the following steps.*Step 1. *We show that for all and all and so , , , , , and are bounded.

Indeed, let . Then, noting that

we have
which yields that
Using an induction, we obtain
for all . Hence is bounded, and so are , , , , and .*Step 2. *We show that and . Indeed, it follows from condition (C1) and (C2) that
Also from , we get
*Step 3. *We show that . To this end, set for . Then it follows from (C1) and (C3) that
Define
Observe that
It follows from (3.13) that
Since and are bounded, by (C1), (3.10), (3.11), and (3.14) we obtain that
Hence by Lemma 2.2, we have
It follows from (3.11) and (3.12) that
*Step 4. *We show that . In fact, from (IS) it follows that
So we have
Thus, from condition (C3), Steps 2, and 3, we have
*Step 5. *We show that . To prove this, let a subsequence of be such that

and for some . From Step 4, it follows that .

Now let , where . Then we can write

Putting
by Step 4 and using Lemma 2.1, we obtain
The last inequality implies
It follows that
where is a constant such that for all and . Taking the as in (3.26) and noticing the fact that the two limits are interchangeable due to the fact that is uniformly continuous on bounded subsets of from the strong topology of to the topology of , we have
Indeed, letting , from (3.26) we have
So, for any , there exists a positive number such that for any ,
Moreover, since as , the set is bounded and the duality mapping is uniformly continuous on bounded subset of , there exists such that, for any ,
Choose , we have for all and ,
which implies that
Since , we have
Since is arbitrary, we obtain that
*Step 6. *We show that . By using (IS), we have
Applying Lemma 2.1 and (3.6), we obtain
It then follows that
where . Put
From the condition (C1) and Step 5, it follows that , , and . Since (3.37) reduces to
from Lemma 2.3 with , we conclude that . This completes the proof.

Corollary 3.3. *Let ** be a uniformly smooth Banach space. Let *, ,, , , , , , , *and ** be the same as in Theorem 3.2. Then the conclusion of Theorem 3.2 still holds.*

*Proof. *Since is a uniformly smooth Banach space, is reflexive, the norm is uniformly Gâteaux differentiable, and every nonempty weakly compact convex subset of has the fixed point property for nonexpansive mappings. Thus the conclusion of Corollary 3.3 follows from Theorem 3.2 immediately.

Corollary 3.4. *Let ** be a nonempty closed convex subset of a uniformly smooth Banach space **. Let ** be a nonexpansive mapping with **. Let **, **, and ** be three sequences in ** which satisfy the control conditions * (C1)–(C3)* in Theorem 3.2. Then for the initial guess ** and **, the sequence ** generated by * (1.6) *converges strongly to a fixed point of ** under the assumption **.*

*Remark 3.5. *(1) In general, the condition (C3) in Theorem 3.2 and the condition of Qin et al. [16, Theorem 2.1] are not comparable; neither of them implies the other. Theorem 3.2 (and Corollary 3.4) removes the conditions and imposed on the control parameters and of Qin et al. [16, Theorem 2.1].

(2) Theorem 3.2 (and Corollary 3.3) complements the corresponding results in Moudafi [11], Xu [12], and Yao et al. [15]. In particular, if in (IS), then (IS) in Theorem 3.2 reduces a new one for finding a fixed point of :

(3) Corollary 3.4 with in (IS) develops the corresponding results of Shioji and Takahashi [8], Wittmann [9] without the condition as well as the result of Chang [2] in which the condition was assumed.

Next, we consider the viscosity iterative scheme with the weakly contractive mappings instead of the contractions.

Theorem 3.6. *Let be a reflexive Banach space having a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of has the fixed point property for nonexpansive mappings. Let be a nonempty closed convex subset of and let be a nonexpansive mapping from into itself with . Let ,and be sequences in which satisfy the conditions (C1)–(C3) in Theorem 3.2. Let be a weakly contractive mapping and let be chosen arbitrarily. Letbe the sequence generated by
**
If , then converges strongly to , where is a sunny nonexpansive retraction from onto .*

*Proof. *It follows from Remark 3.1 that is the sunny nonexpansive retract of . Denote by the sunny nonexpansive retraction of onto . Then is a weakly contractive mapping of into itself. Indeed,
Lemma 2.4 assures that there exists a unique element such that . Such a is an element of .

Now we define an iterative scheme as follows:
Let be the sequence generated by (3.37). Then, by taking as in , Theorem 3.2 with a constant assures that converges strongly to as . For any , observe that
Then we have
Thus, for , we obtain the following recursive inequality:
Since , it follows from Lemma 2.5 that . Hence
This completes the proof.

Corollary 3.7. *
Let
be a uniformly smooth Banach space. Let
, , , , , , , , , and
be the same as in Theorem 3.6. Then the conclusion of Theorem 3.6 still holds.
*

*Remark 3.8. *(1) Theorem 3.6 (and Corollary 3.7) develops and complements the corresponding results in Moudafi [11], Qin et al. [16], Shioji and Takahashi [8], Wittmann [9], Xu [10, 12], and Yao et al. [15].

(2) Even in the case of in Theorem 3.6, Theorem 3.6 appears to be independent of Theorem 5.6 of Wong et al. [13] in which the control conditions (C1) and were utilized. In fact, it appears to be unknown whether a reflexive and strictly convex space satisfies the fixed point property for nonexpansive mappings.

(3) The merits of our results in this paper are that fewer restrictions are imposed on the control parameters , , and . All of our results can be viewed as a supplement to the results obtained by Qin et al. [16], Kim and Xu [14], and Yao et al. [15].

(4) The conclusions of Theorems 3.2 and 3.6 still hold if is assumed to be strictly convex instead of having the fixed point property for nonexpansive mappings.

#### Acknowledgments

The author thanks the editor Professor Simeon Reich and the referees for their careful reading and valuable suggestions to improve the writing of this paper.