#### Abstract

In this article, we suggest some and strong convergence results of a three-step Sahu–Thakur iteration process for Garcia-Falset maps in the nonlinear setting of CAT(0) spaces. We furnish a new example of Garcia-Falset maps and prove that its three-step Sahu–Thakur iterative process is more effective than the many well-known iterative processes. Our results improve and extend some recently announced results of the current literature.

#### 1. Introduction

Let be a metric space and . A self-map of is called a contraction map if and only if for every choice of two points , there is a real constant such that . The self-map is regarded as nonexpansive if and only if . A point, namely, , is called a fixed point for the self-map of if and only if , and the notation will denote the set of all fixed points. In 1922, Banach  observed that every self-contraction of a closed subset of a Banach space admits a unique fixed point, namely, , and for every choice of initial value, Picard iterative method (sometimes called the successive approximation method) converges to this . In 1930, Caccioppoli  quickly noted that the Banach result is valid in the setting of general complete metric spaces. The existence of fixed points for a nonexpansive map was independently established by Browder  and Göhde  in the year 1965. They observed that if is closed convex and bounded in a uniformly convex Banach space, then has a fixed point (may not unique). In 2003, Kirk  obtained this result in the nonlinear setting of CAT(0) spaces. Every nonexpansive map is not necessary to be contraction. Hence, the study of fixed points of nonexpansive maps is more harder but more important than the corresponding study of contraction maps. In 2008, Suzuki  suggested a very weaker concept of nonexpansive maps. A self-map of a subset of a metric space is regarded as a Suzuki map (sometimes called the Suzuki (C)-map) if for any choice of two points , one has

Suzuki  proved many interesting properties of these maps. He proved that any nonexpansive map is Suzuki, but the inverse is not necessary to be valid, in general. He also generalized the Browder–Gohde result to this new setting of maps. In , Nanjaras et al. extended and improved all of the results of Suzuki  to the nonlinear setting of CAT(0) spaces. After this, many papers appeared on the topic of Suzuki maps in linear and nonlinear spaces (cf.  and references therein).

Keeping Suzuki (C)-maps in mind, García-Falset et al.  suggested the notion of (E)-maps as follows: a self-map of a subset of a metric space is regarded as a Garcia-Falset map (sometimes called the Garcia-Falset (E)-map) if for any choice of two points , one has

Since by Suzuki , every Suzuki map of a nonempty subset of a metric space satisfies for every two elements . It immediately follows that every Suzuki map is a Garcia-Falset map having real constant . Interestingly, in this article, we will show that not every Garcia-Falset map is a Suzuki map. Thus, we conclude that the class of Garica-Falset maps properly contains the corresponding class of Suzuki maps. García-Falset et al.  established many fixed-point existence results for this larger class of maps.

The existence of fixed points for a self-map when established then to approximate such a fixed point via a most suitable iterative process is not an easy research work. As many know, the unique fixed point of a contraction map can be approximated by using the simplest Picard iterative process . We also know that the fixed point of a nonexpansive map cannot be approximated by using the Picard iterative process, in general. Thus, to obtain a relatively effective rate of convergence and to overcome such type of difficulties, many new iterative processes are designed by authors (see, e.g., Mann , Ishikawa , Agarwal et al. , Noor , Abbas and Nazir , and Thakur et al. ). Among these iterative processes, Sahu et al.  and Thakur et al.  independently suggested an effective three-step iterative scheme for approximating fixed points of nonexpansive maps:where and denotes the set of all positive integers.

In , Thakur et al. established many interesting weak and strong convergence results for the class of nonexpansive maps under iterative process (3) in the ground setting of Banach spaces. Recently in 2019, Usurelu and Postolache  improved and extended the main outcome of Thakur et al.  to the larger class of Suzuki maps. Very recently in 2020, Usurelu et al.  quickly improved and extended the main outcome of Usurelu and Postolache  to the general setting of Garcia-Falset maps. The aim of this research work is to improve and extend the main outcome of Usurelu et al.  to the nonlinear setting of CAT(0) spaces. Simultaneously, our results improve and extend the corresponding results of Thakur et al.  and Usurelu and Postolache . Moreover, we suggest a new example of Garcia-Falset maps which exceeds the corresponding class of Suzuki maps and prove that its Sahu–Thakur iterative process (3) is more effective than the many other iterative schemes.

#### 2. Preliminaries

Take a closed interval in the set , and suppose are any elements of a metric space . Then, a map is called geodesic provided that , , and for every choice of two points . In this case, we regard the image of as a geodesic segment combining the elements and , and it is denoted simply by whenever it is unique. Recall that a metric space is known as a geodesic space if any two points in are combined by a geodesic. is called uniquely geodesic in the case when every two elements , one can find a unique geodesic combining them. A subset of S is called convex whenever any two points are given, one is able to combine them by a geodesic in . Notice that the image of each such geodesic always contains in the set .

Now, a given geodesic triangle in a geodesic metric space is composed of three elements in and a choice of three geodesic segments combining them.

A triangle in the Euclidean plane is called a comparison triangle for a given geodesic triangle provided that

An element is known as a comparison element for provided that . Comparison elements on and can be stated in the similar way.

Definition 1. Suppose is a given geodesic triangle in a metric space . We say that is endowed with the CAT(0) inequality whenever for and for their comparison elements, namely, , it follows thatNotice that a geodesic space S is called a CAT(0) space provided that each of its geodesic triangle is endowed with the CAT(0) inequality. Some other equivalent definition of CAT(0) can be found in . Moreover, each CAT(0) space is uniquely geodesic. Some well-known examples of CAT(0) spaces include pre-Hilbert spaces, metric trees, and Euclidean buildings. For more details and the literature of fixed-point theory in nonlinear spaces, we refer the reader to .
We now state a result from .

Lemma 1. Suppose is a given CAT(0) space.(i)If and is fixed, then there exists a unique element, namely, , such thatWe shall normally represent by the unique element satisfying (6).(ii)If and is fixed, thenAssume that is convex and closed in a CAT(0) space and is bounded. Fix , and setWe normally denote and define the asymptotic radius of the sequence wrt asWe normally denote and define the asymptotic center of the sequence wrt asIt is known that the asymptotic center is simply denoted by wrt . Interestingly, the set contains unique elements in the setting of complete CAT(0) spaces (see ).

Definition 2. A bounded sequence of elements of a CAT(0) space is called -convergent to provided that is the unique asymptotic center for every subsequence of . When is of , then we write .
Every CAT(0) space enjoys the Opial-like property , that is, if is -convergent to , then for every choice of , it follows that

Lemma 2. (see ). Suppose is a CAT(0) space and is bounded. Then, always has a -convergent subsequence.

Lemma 3. (see ). Suppose is a CAT(0) space, is closed and convex, and is bounded. Then, the asymptotic center of the sequence is always in .

The following characterizations of Garcia-Falset maps can be found in .

Proposition 1. Let be a nonempty subset of a complete CAT(0) space and .(i)If is a Suzuki mapping, then is also a Garcia-Falset map(ii)If is a Garcia-Falset map having , then for every choice of and , one has (iii)If is a Garcia-Falset map, then the set is closedWe know the following characterization of the CAT(0) space from .

Lemma 4. Assume that is a given complete CAT(0) space, and fix . Suppose and , are two sequences of elements of such that , , and for some real number ; then, .

#### 3. Convergence Theorems in CAT(0) Spaces

The aim of this section is to establish and strong convergence of the Sahu–Thakur iterative scheme for Garcia-Falset maps. Throughout the section, we shall denote simply by S a complete CAT(0) space. Notice that Sahu–Thakur iterative process (3) in the framework of CAT(0) spaces is written as follows:where .

We first establish a key lemma as follows.

Lemma 5. Suppose be any nonempty closed convex subset of , and assume that is a Garcia-Falset map having . If the sequence is defined by equation (12), then exists for each choice of .

Proof. Suppose . Keeping Proposition 1 (ii) in mind, one hasUsing the above inequalities, we haveHence, the real sequence is nonincreasing and bounded. Thus, we conclude that exists for each choice of .
Using Lemma 5 and the definition of the Garcia-Falset map, we establish the following result in CAT(0) spaces.

Theorem 1. Let be a nonempty convex closed subset of S, and let be a Garcia-Falset operator. For arbitrary chosen , let the sequence be defined by equation (12). Then, if and only if is bounded and .

Proof. Suppose that the sequence is bounded and . Fix an element . We are going to show that . Since is a Garcia-Falset map,Hence, it is seen that , but is singleton, and one can conclude that . Thus, is the element of .
Conversely, let the set . We are going to show that is bounded and . Fix ; then, by Lemma 5, exists, and is bounded. PutThe proof of Lemma 5 suggests thatBy using Proposition 1 (ii), one hasAlso, the proof of Lemma 5 suggests thatFrom equations (17) and (19), we obtainFrom equation (20), we haveApplying Lemma 4, we obtainThis completes the proof.
We now establish a strong convergence for Garcia-Falset mappings on compact domains in CAT(0) spaces.

Theorem 2. Assume that is a nonempty convex compact subset of . Suppose and are as given in Theorem 1 and . Then, converges strongly to a point of .

Proof. By the compactness property of the set , we can construct a subsequence of such that , for some . Since the self-map is a Garcia-Falset map, there exists a real constant , withIn the view of Theorem 1, . Now, using and , we have from equation (23) , but is a metric space, so the limit of is unique, that is, . Hence, . By Lemma 5, exists. Hence, is the strong limit of .

Theorem 3. Assume that is a nonempty closed convex subset of . Suppose and are as given in Theorem 1. If and , then converges strongly to a point of .

Proof. Since the proof is elementary, we skip it.
We now give the complete definition of condition I, which was originally introduced by Sentor and Dotson in .

Definition 3. Assume that is a nonempty subset of . One says that a self-map of has condition whenever there is a nondecreasing map which satisfies if and only if and for every choice of and for each choice of .
Now, we state and prove a strong convergence result based on condition I.

Theorem 4. Assume that is a nonempty closed convex subset of . Suppose and are as given in Theorem 1 and . Then, converges strongly to a point of if has condition I.

Proof. The conclusions of Theorem 1 provide usCondition I of G suggestsUsing equation (24) and condition I, one hasHowever, the map is nondecreasing with and for each choice of , soThe conclusions of Theorem 3 suggest that is strongly convergent in the set .
Finally, we suggest -convergence for maps having (E)-condition.

Theorem 5. Assume that is a nonempty closed convex subset of . Suppose and are as given in Theorem 1 and . Then, converges to a point of .

Proof. By Theorem 1, one can conclude that is bounded and is such that . We may suppose that , for every subsequence of . The next purpose is to obtain . If we select any , then there exists a subsequence, namely, , of with the property . Using Lemmas 2 and 3, we may conclude that a subsequence of exists having some -limit in , but Theorem 1 suggests that , and also, is endowed with the (E)-condition, so we haveBy considering lim sup on both sides of equation (28), we obtain . Lemma 5 suggests that exists. We need to show that . Assume, on the contrary, that is different from . So, by uniqueness of asymptotic centers, one hasConsequently, . This is a contradiction, and so, one can conclude that and .
It is now remaining to prove is a -convergent sequence in the set . For this purpose, we need which is a singleton set. If is any subsequence of , then by Lemmas 2 and 3, we can choose the -convergent subsequence of having some in . Assume that and . We have already showed that and . We are going to prove that . Suppose not, then exists, and also, the asymptotic centers are singleton, and we havewhich is a contradiction. Thus, . Hence, . This completes the proof.

#### 4. Numerical Example

We now furnish a new numerical example of Garcia-Falset maps and show that its Sahu–Thakur iterative scheme is more effective than the other well-known existing iterative schemes.

Example 1. We may set a self-map of by the rule for and for . To observe that the map is a Garcia-Falset map, we will establish for every choice of and some . We choose and divide the proof in parts as follows: Let ; then, . Then, Let , so and . Then, Let and , so and . Then, Let and , so and . Then,The cases discussed above show that is a Garcia-Falset map on the set . By choosing and , we may observe that and , that is, is not a Suzuki map on . We now choose and and for every natural number . In Table 1, strong convergence of Sahu–Thakur iteration (ST) [20, 21] is compared with the other existing iterative schemes such as Picard , Mann , Ishikawa , Noor , Agarwal et al. et al. , Abbas and Nazir , and Thakur et al. . Both Table 1 and Figure 1 well suggest the effectiveness of Sahu–Thakur iteration in the class of Garcia-Falset maps.

#### 5. Conclusion

The extension of fixed-point results from the context of the linear domain to the general context of the nonlinear domain has its own significance. In , Takahashi suggested the notion of convexity in metric spaces and studied some fixed-point results in this context. After this, many different convexity structures in a context of metric spaces are introduced. In , Usurelu et al. established some convergence results for Garcia-Falset maps under Sahu–Thakur in the linear context of Banach spaces. In this paper, we have extended their results to the nonlinear context of CAT(0) spaces. We have also suggested a new example of Garcia-Falset and showed that its Sahu–Thakur iterative process is more effective than the other prominent iterative schemes. Keeping the previous discussion in mind, the cases when the map is Suzuki or else is nonexpansive are now special cases of our main results. Thus, our results simultaneously improve and extend the corresponding results in [79, 11, 1822, 37] from the setting of nonexpansive and Suzuki maps to the setting of Garcia-Falset maps and from the linear domain to the general setting of the nonlinear domain .

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.