#### Abstract

Let a reflexive Banach space having a uniformly Gâteaux differentiable norm. Let be a nonempty closed convex subset of , a continuous pseudocontractive mapping with , and a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant . Let and be sequences in satisfying suitable conditions and for arbitrary initial value , let the sequence be generated by If either every weakly compact convex subset of has the fixed point property for nonexpansive mappings or is strictly convex, then converges strongly to a fixed point of , which solves a certain variational inequality related to .

#### 1. Introduction and Preliminaries

Throughout this paper, we denote by the norm and a real Banach space and the dual space of , respectively. Let be a nonempty closed convex subset of . For the mapping , we denote the fixed point set of by ; that is, .

Let denote the normalized duality mapping from into defined by
where denotes the generalized duality pair between and . Recall that the norm of is said to be *Gâteaux differentiable* if
exists for each in its unit sphere . Such an is called a *smooth* Banach space. 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) is attained uniformly for . It is known that is smooth if and only if the normalized duality mapping is single valued. It is well known that if is uniformly smooth, then the duality mapping is norm-to-norm uniformly continuous on bounded subsets of , and that if has a uniformly Gâteaux differentiable norm, then is norm-to-weak* uniformly continuous on each bounded subsets of [1, 2].

It is relevant to the results of this paper to note that while every uniformly smooth Banach space is a reflexive Banach space having a uniformly Gâteaux differentiable norm, the converse does not hold. To see this, consider to be the direct sum , the class of all those sequences with and (see [3]). If for , where either or , then is a reflexive Banach space with a uniformly Gâteaux differentiable norm but is not uniformly smooth (see [3–5]). We also observe that the spaces which enjoy the fixed point property (shortly, F.P.P) for nonexpansive mappings are not necessarily spaces having a uniformly Gâteaux differentiable norm. On the other hand, the converse of this fact appears to be unknown as well.

A Banach space is said to be *strictly convex* if
A Banach space is said to be *uniformly convex* if for all , where is the *modulus of convexity* of defined by
It is well known that a uniformly convex Banach space is reflexive and strictly convex [1] and satisfies the F.P.P. for nonexpansive mappings. However, it appears to be unknown whether a reflexive and strictly convex space satisfies the F.P.P. for nonexpansive mappings.

Recall that a mapping with domain and range in is called *pseudocontractive* if the inequality
holds for each, and for all . From a result of Kato [6], we know that (5) is equivalent to (6) below; there exists such that
for all . The mapping is said to be *strongly pseudocontractive* it there exists a constant and such that
for all .

The class of pseudocontractive mappings is one of the most important classes of mappings in nonlinear analysis and it has been attracting mathematician’s interest. In addition to generalizing the nonexpansive mappings (the mappings for which , for all ), the pseudocontractive ones are characterized by the fact that is pseudocontractive if and only if is accretive, where a mapping with domain and range in is called accretive if the inequality holds for every and for all .

Within the past 40 years or so, many authors have been devoting their study to the existence of zeros of accretive mappings or fixed points of pseudocontractive mappings and iterative construction of zeros of accretive mappings and of fixed points of pseudocontractive mappings (see [5, 7–10]). Also, several iterative methods for approximating fixed points (zeros) of nonexpansive and pseudocontractive mappings (accretive mappings) in Hilbert spaces and Banach spaces have been introduced and studied by many authors. We can refer to [11–17] and the references in therein.

In 2007, Rafiq [15] introduced a Mann-type implicit iterative method (9) for a hemicontractive mapping as where is a compact convex subset of a real Hilbert space and for some , and proved that converges strongly to a fixed point of .

In 2007, Yao et al. [16] introduced an iterative method (10) below for approximating fixed points of a continuous pseudocontractive mapping without compactness assumption on its domain in a uniformly smooth Banach space: for arbitrary initial value and a fixed anchor , where , , and are three sequences in satisfying some appropriate conditions. By using the Reich inequality [9] in uniformly smooth Banach spaces where is a nondecreasing continuous function, they proved that the sequence generated by (10) converges strongly to a fixed point of . In particular, in 2007, by using the viscosity iterative method studied by [18, 19], Song and Chen [17] introduced a modified implicit iterative method (12) below for a continuous pseudocontractive mapping without compactness assumption on its domain in a real reflexive and strictly convex Banach space having a uniformly Gâteaux differentiable norm: for arbitrary initial value , where and are two sequences in satisfying some appropriate conditions and is a contractive mapping, and proved that the sequence generated by (12) converges strongly to a fixed point of , which is the unique solution of a certain variational inequality related to .

In this paper, inspired and motivated by the above-mentioned results, we introduce the following the iterative method for a continuous pseudocontractive mapping : for arbitrary initial value , where and are two sequences in and is a bounded continuous strongly pseudocontractive mapping with a pseudocontractive constant . Whether a reflexive Banach space has a uniformly Gâteaux differentiable norm such that every weakly compact convex subset of has the fixed point property for nonexpansive mappings, or a reflexive and strict convex Banach space has a uniformly Gâteaux differentiable norm, we establish the strong convergence of the sequence generated by proposed iterative method (13) to a fixed point of the mapping, which solves a certain variational inequality related to . The main result generalizes, improves, and develops the corresponding results of Yao et al. [16] and Song and Chen [17] as well as Rafiq [15].

We need the following well-known lemmas for the proof of our main result.

Lemma 1 (see [1, 2]). *Let be a Banach space and let be the normalized duality mapping on . Then for any , the following inequality holds:
*

Lemma 2 (see [20]). *Let be a sequence of nonnegative real numbers satisfying
**
where and satisfy the following conditions: *(i)* and or, equivalently, ,*(ii)* or .**
Then . *

#### 2. Iterative Methods

We need the following result which was given in [10].

Proposition 3. *Let be a closed convex subset of a Banach space . Suppose that are two continuous mappings from into itself, which are pseudocontractive and strongly pseudocontractive, respectively. Then there exists a unique path , , satisfying
**
Further, the following hold.*(i)*Suppose that there exists a bounded sequence ** in ** such that **, while ** is bounded. Then the path ** is bounded. *(ii)*In particular, if ** has a fixed point in **, then the path ** is bounded. *(iii)*If ** is a fixed point of **, there exists ** such that *

We prepare the following result for the existence of a solution of the variational inequality related to .

Theorem 4. *Let be a nonempty closed convex subset of a Banach space and let be a continuous pseudocontractive mapping from into itself with and let be a continuous bounded strongly pseudocontractive mapping with a pseudocontractive coefficient . For each , let be defined by
**
If one of the following assumptions holds: *(H1)* is a reflexive Banach space, the norm of is uniformly Gâteaux differentiable, and every weakly compact convex subset of has the fixed point property for nonexpansive mappings; *(H2)* is a reflexive and strictly convex Banach space and the norm of is uniformly Gâteaux differentiable, **
then the path converges strongly to a point in , which is the unique solution of the variational inequality
*

*Proof. *In case of (H1), the result follows from Theorem 2 of [10]. So, we prove only the case of (H2). We follow the method of proof in [5, 10].

By Proposition 3, the path exists. It remains to show that it converges strongly to a fixed point of as . As a consequence of Theorem 6 of [21], the mapping has a nonexpansive inverse, denoted by , which maps into itself with . By Proposition 3(ii), is bounded. Since is a bounded mapping, is bounded. From (18), we have
and so is bounded (as ). Since
we derive that
Let be a sequence in such that as and let . Since is bounded, we may define by . Since is reflexive, as , and is continuous and convex, and the set
is a nonempty (due to Theorem 1.2 of [22]). Since , let . Then, since is strict convex, the set
is singleton. Let for some . We also know that and
Therefore, . Since is pseudocontractive and is a fixed point of , we derive from Proposition 3(iii) that
Now let and . Then by Lemma 1, we have
Let . Then by the assumption on , there exists such that and
for all . Consequently,
Therefore, we may choose a subsequence of such that
From (26) and the fact that is strongly pseudocontractive, we have
which implies that . Thus, by (30), we conclude that converges strongly to . To prove that actually the net converges strongly to , let be another subsequence of such that , as and , where . Then Proposition 3(iii) implies that and . This implies that
Since is strongly pseudocontractive, and the strong exists. This same argument may be used to conclude that is the only solution of the variational inequality (19). This completes the proof.

Using Theorem 4, we establish our second main result.

Theorem 5. *Let be a Banach space and let be a nonempty closed convex subset of . Let be a continuous pseudocontractive mapping such that , and let be a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant . Let and be sequences in satisfying the following conditions:*(C1)* and ;*(C2)*. **
For arbitrary initial value , let the sequence be defined by
**
If one of the following assumptions holds: *(H1)* is a reflexive Banach space, the norm of is uniformly Gâteaux differentiable, and every weakly compact convex subset of has the fixed point property for nonexpansive mappings; *(H2)* is a reflexive and strictly convex Banach space and the norm of is uniformly Gâteaux differentiable, **
then converges strongly to a fixed point of , which is the unique solution of the variational inequality
*

*Proof. *We divide the proof into several steps as follows.*Step 1*. We show that is bounded. To this end, let . Then, noting that
we have
which implies
So, we obtain
By induction, we have
Hence, is bounded. Since is a bounded mapping, is bounded. From (33), it follows that
and so is bounded (as ). *Step 2*. We show that . In fact, by (33) and the condition (C1), we have
*Step 3.* We show that
where with being defined by . To this end, we note that
Then, it follows that
which implies that
From Proposition 3, we know that , and , are bounded. Since and are also bounded and by Step 2, taking the upper limit as in (45), we get
where is a constant such that for all and . Taking the as in (46) and noticing the fact that the two limits are interchangeable due to the fact that is norm-to-weak* uniformly continuous on each bounded subsets of , we have
*Step 4.* We show that , where with being defined by and is the unique solution of the variational inequality (34) by Theorem 4. First, from (33) and (35), we have
This implies that
where and . We observe that and . From the condition (C2) and Step 3, it is easily seen that and . Thus, applying Lemma 2 to (49), we conclude that . This completes the proof.

Corollary 6. *Let be a uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a continuous pseudocontractive mapping such that and let be a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant . Let and be two sequences in satisfying the conditions (C1) and (C2) in Theorem 5. For arbitrary initial value , let the sequence be generated by (33) in Theorem 5. Then converges strongly to a fixed point of , which is the unique solution of the variational inequality (34).*

Corollary 7 (see [16, Theorem 3.1]). *Let be a uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a continuous pseudocontractive mapping such that . Let , and be three sequences in satisfying the conditions (C1) and (C2) in Theorem 5 and for . For arbitrary initial value and a fixed anchor , let the sequence be generated by
**
Then converges strongly to a fixed point of , which is the unique solution of the variational inequality
*

*Proof. *Taking , for all as a constant function, the result follows from Corollary 6.

Corollary 8. *Let be a uniformly convex Banach space has a uniformly Gâteaux differentiable norm and let be a nonempty closed convex subset of . Let be a continuous pseudocontractive mapping such that and let be a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant . Let and be two sequences in satisfying the conditions (C1) and (C2) in Theorem 5. For arbitrary initial value , let the sequence be generated by (33) in Theorem 5. Then converges strongly to a fixed point of , which is the unique solution of the variational inequality (34). *

*Remark 9. * (1) Theorem 5 extends and improves Theorem 3.1 of Yao et al. [16] in the following aspects.(a) is replaced by a continuous bounded strongly pseudocontractive mapping . (b) The uniformly smooth Banach space is extended to a reflexive Banach space having a uniformly Gâteaux differentiable norm. (c) The condition in [16] is weakened to and as . (2)It is worth pointing out that in Corollaries 6 and 7, we do not use the Reich inequality (11) in comparison with Theorem 3.1 of Yao et al. [16].(3)Theorem 5 and Corollary 8 also develop and complement Theorem 3.1 and Corollary 3.2 of Song and Chen [17] by replacing the contractive mapping with a continuous bounded strongly pseudocontractive mapping in the iterative scheme in [17].(4)The assumption (H1) in Theorems 4 and 5 appears to be independent of the assumption (H2).(5)We point out that the results in this paper apply to all spaces, .

#### Acknowledgment

This study was supported by research funds from Dong-A University.