#### Abstract

In this article, we prove some weak and strong convergence theorems for mappings satisfying condition (*E*) using the iterative scheme in the setting of Banach spaces. We offer a new example of mapping with condition (*E*) in support of our main result. Our results extend and improve many well-known corresponding results of the current literature.

#### 1. Introduction

Let be a Banach space, and . An element is called a fixed point for if . We denote by the set of all fixed points of the map . Throughout the work, we will denote by the set of all natural numbers. When is nonexpansive, that is, for all , one has , then is nonempty provided that is uniformly convex and is convex closed bounded (see [1–3] and others). In 2008, Suzuki [4] introduced a new class of nonlinear mappings, which is the generalization of the class of nonexpansive mappings. A mapping is said to satisfy condition (*C*) (or Suzuki mapping) if for all ,

In 2011, Garcia-Falset et al. [5] extended condition (*C*) to the general formulations as follows. A mapping is said to satisfy condition if there exists some such that

A mapping is said to satisfy condition (*E*) (or Garcia-Falset mapping) whenever satisfies condition for some . Garcia-Falset et al. [5] proved that every Suzuki mapping satisfies condition (*E*) with . Notice also that the class of Garcia-Falset mappings also includes many other classes of generalized nonexpansive (see [6] for details). In this paper, we study in deep this general class of mapping.

The iterative approximation of fixed points for nonlinear operators is an active research area nowadays (see, e.g., [7–11] and others). The Banach contraction principle suggests a Picard iterative scheme for finding the unique fixed point of a given contraction mapping. However, the Picard iterative scheme does not always converge to a fixed point of a nonexpansive mapping. Thus, to overcome such difficulties and to obtain a better speed of convergence, many iterative schemes are available in the literature. Let be a nonempty convex subset of a Banach space and . Assume that for all . Then, the well-known Picard, Mann [12], Ishikawa [13], Noor [14], Agarwal [15], Abbas and Nazir [16], Thakur [17], [18], and AK [19] iterative schemes are, respectively, read as follows:

In [15], Agarwal et al. proved that the iterative scheme (7) converges faster than the iterative schemes (3)–(5) for contraction mappings. In [17], Thakur et al. with the help of a numerical example proved that the iterative scheme (9) converges faster than all of the iterative schemes (3)–(8) for the general setting of Suzuki mappings. In [18], Ullah and Arshad with the help of a numerical example proved that the iterative scheme (10) is better than the leading iterative scheme (9) for Suzuki mappings in Banach spaces. In [19], the authors proved that the AK iterative scheme (11) is stable and converges faster than many well-known iterative schemes for contraction mappings. The purpose of this research is to study the iterative scheme (11) for the generalized class of Garcia-Falset mappings. We also give a new example of the Garcia-Falset mapping and show that its iteration process is more efficient than all of the above schemes.

#### 2. Preliminaries

Let be any nonempty subset of a Banach space and let be a bounded sequence in . For , we set

The asymptotic radius of relative to is given by

The asymptotic center of relative to is the set

When the space is uniformly convex [20], then the set is singleton. Notice also that the set is convex as well as nonempty provided that is weakly compact convex (see, e.g., [21, 22]).

We say that a Banach space has Opial’s property [23] if and only if for all in which weakly converges to and for every -, one has

The following lemma gives many examples of Garcia-Falset mappings.

Lemma 1. *(see [5]). Let be a mapping on a subset of a Banach space. If satisfies condition ( C), then also satisfies condition (E) with .*

Lemma 2. *(see [5]). Let be a mapping on a subset of a Banach space. If satisfies condition ( E), then for all and , we have .*

Lemma 3. *(see [5]). Let be a mapping on a subset of a Banach space having the Opial property. Assume that satisfies the condition ( E). If converges weakly to and , then .*

In 1991, Schu [24] proved the following useful fact.

Lemma 4. *Let be a uniformly convex Banach space and for all . If and are two sequences in such that , , and for some , then .*

#### 3. Convergence Theorems in Uniformly Convex Banach Spaces

In this section, we shall state and prove our main results. First, we give the following key lemma.

Lemma 5. *Let be a nonempty closed convex subset of a Banach space and let be a mapping satisfying condition ( E) with . Let the sequence be defined by (11), then exists for all .*

*Proof. *Suppose . By Lemma 2, we havewhich implies thatThus, is bounded and nonincreasing, which implies that exists for each .

The following theorem will be used in the upcoming results.

Theorem 1. *Let be a nonempty closed convex subset of a uniformly convex Banach space and let be a mapping satisfying condition ( E). Let be the sequence defined by (11). Then, if and only if is bounded and .*

*Proof. *Let be bounded and . Let . We shall prove that . Since satisfies condition (*E*), we haveIt follows that . Since is a singleton set, we have . Hence, .

Conversely, we assume that and . We shall prove that is bounded and . By Lemma 5, exists and is bounded. PutBy (16), we haveBy Lemma 2, we haveBy (17), we haveSo, we can getFrom (20) and (24), we getUsing (19) and (25), we haveHence,Applying Lemma 4, we obtainFirst, we discuss the strong convergence of defined by (11) for mappings with condition (*E*).

Theorem 2. *Let be a nonempty convex compact subset of a uniformly convex Banach space and let and be as in Theorem 1 and . Then, converges strongly to a fixed point of .*

*Proof. *By Theorem 1, . By compactness of , we can find a subsequence of such that converges strongly to for some . Since satisfies condition (*E*), there exists some , such thatLetting , we get . By Lemma 5, exists. Hence, is the strong limit of .

Proof of the following result is elementary and hence omitted.

Theorem 3. *Let be a nonempty closed convex subset of a uniformly convex Banach space and let and be as in Theorem 1. If and , then converges strongly to a fixed point of .*

Now, we establish a strong convergence result for Garcia-Falset mappings using the iteration process with the help of condition (*I*).

*Definition 1. *(see [25]). Let be a nonempty subset of a Banach space . A mapping is said to satisfy condition (*I*) if there is a function satisfying and for all such that for all .

Theorem 4. *Let be a nonempty closed convex subset of a uniformly convex Banach space and let and be as in Theorem 1 and . If satisfies condition ( I), then converges strongly to a fixed point of .*

*Proof. *From Theorem 1, it follows thatSince satisfies condition (*I*), we haveFrom (30), we get is a nondecreasing function with and for each . Hence,The conclusion follows from Theorem 3.

Finally, we establish a weak convergence of for mappings with condition (*E*).

Theorem 5. *Let be a uniformly Banach space with the Opial property, a nonempty closed convex subset of , and let and be as in Theorem 1 and . Then, converges weakly to a fixed point of .*

*Proof. *By Theorem 1, is bounded and . By the Milman–Pettis theorem, the space is reflexive. Thus, by Eberlin’s theorem, there exists a subsequence of such that converges weakly to some . By Lemma 3, we have . It is sufficient to show that converges weakly to . In fact, if does not converge weakly to . Then, there exists a subsequence of and such that converges weakly to and . Again by Lemma 3, . By Lemma 5 together with the Opial property, we haveThis is a contradiction. Hence, the conclusions are reached.

#### 4. Numerical Example

In this section, we present a new example of the Garcia-Falset mapping which is not a Suzuki mapping. Using this example, we compare the rate of convergence of the iterative scheme with the other iterative schemes. This example also shows that the converse of Lemma 1 may not hold in general.

*Example 1. *Let . Set on as follows:Let . We shall prove that for all . Case (i): if , then . Now, Case (ii): if , then and . Now, Case (iii): if and , then and . Now,From the above cases, we conclude that satisfies condition (*E*). Now choose and . Then, but . Hence, is not a Suzuki mapping. For all , let , , and . Table 1 and Figure 1 show that the iteration scheme converges faster to the fixed point of the mapping as compared with the other known iterative schemes.

#### 5. Conclusions

We have proved several strong and weak convergence results for mappings with condition (*E*) (Garcia-Falset mappings) in the context of Banach spaces. In view of the above discussion, the results for an operator satisfying condition (*C*) or else nonexpansive are special cases of our new results. Hence, our results are more general than the results of Ullah and Arshad [18], Abbas and Nazir [16], Thakur et al. [17], Phuengrattana [26], and many others. Moreover, our results extend the idea of Ullah and Arshad [19] from the setting of contraction mappings to the general setting of Garcia-Falset mappings.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

#### Acknowledgments

The first author and the last author would like to thank Prince Sultan University for funding this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) (group number RG-DES-2017-01-17).