Fixed-Point Convergence Results of a Three-Step Iterative Process in CAT(0) Spaces

In this article, we suggest some Δ and strong convergence results of a three-step Sahu–,akur 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–,akur 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.


Introduction
Let (S, d) be a metric space and ∅ ≠ B⊆S. A self-map G of B is called a contraction map if and only if for every choice of two points b, b ′ ∈ B, there is a real constant 0 ≤ r < 1 such that d(Gb, Gb ′ ) ≤ r d(b, b ′ ). e self-map G is regarded as nonexpansive if and only if d(Gb, Gb ′ ) ≤ d(b, b ′ ). A point, namely, q 0 ∈ B, is called a fixed point for the self-map G of B if and only if Gq 0 � q 0 , and the notation F G will denote the set of all fixed points. In 1922, Banach [1] observed that every self-contraction G of a closed subset B of a Banach space admits a unique fixed point, namely, q 0 , and for every choice of initial value, Picard iterative method (sometimes called the successive approximation method) converges to this q 0 . In 1930, Caccioppoli [2] quickly noted that the Banach result is valid in the setting of general complete metric spaces. e existence of fixed points for a nonexpansive map was independently established by Browder [3] and Göhde [4] in the year 1965. ey observed that if B is closed convex and bounded in a uniformly convex Banach space, then G has a fixed point (may not unique). In 2003, Kirk [5] 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 [6] suggested a very weaker concept of nonexpansive maps. A self-map G of a subset B of a metric space is regarded as a Suzuki map (sometimes called the Suzuki (C)-map) if for any choice of two points b, b ′ ∈ B, one has Suzuki [6] 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 [7], Nanjaras et al. extended and improved all of the results of Suzuki [6] 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. [8][9][10][11] and references therein).
Keeping Suzuki (C)-maps in mind, García-Falset et al. [12] suggested the notion of (E)-maps as follows: a self-map G of a subset B 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 b, b ′ ∈ B, one has Since by Suzuki [6], every Suzuki map of a nonempty subset B of a metric space satisfies It immediately follows that every Suzuki map is a Garcia-Falset map having real constant μ � 3. Interestingly, in this article, we will show that not every Garcia-Falset map is a Suzuki map. us, we conclude that the class of Garica-Falset maps properly contains the corresponding class of Suzuki maps. García-Falset et al. [12] established many fixed-point existence results for this larger class of maps. e 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 [13]. We also know that the fixed point of a nonexpansive map cannot be approximated by using the Picard iterative process, in general.
In [21], akur 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 [22] improved and extended the main outcome of akur et al. [21] to the larger class of Suzuki maps. Very recently in 2020, Usurelu et al. [23] quickly improved and extended the main outcome of Usurelu and Postolache [22] to the general setting of Garcia-Falset maps. e aim of this research work is to improve and extend the main outcome of Usurelu et al. [23] to the nonlinear setting of CAT(0) spaces. Simultaneously, our results improve and extend the corresponding results of akur et al. [21] and Usurelu and Postolache [22]. Moreover, we suggest a new example of Garcia-Falset maps which exceeds the corresponding class of Suzuki maps and prove that its Sahu-akur iterative process (3) is more effective than the many other iterative schemes.

Preliminaries
Take a closed interval [0, v] in the set R, and suppose b, b ′ are any elements of a metric space S � (S, d).
en, a map g: [0, v] ⟶ S is called geodesic provided that g(0) � b, g(v) � b ′ , and d(g(r 1 ), g(r 2 )) � |r 1 − r 2 | for every choice of two points r 1 , r 2 ∈ [0, v]. In this case, we regard the image of g as a geodesic segment combining the elements b and b ′ , and it is denoted simply by [b, b ′ ] whenever it is unique. Recall that a metric space S is known as a geodesic space if any two points in S are combined by a geodesic. S is called uniquely geodesic in the case when every two elements b, b ′ ∈ S, one can find a unique geodesic combining them. A subset B of S is called convex whenever any two points b, b ′ are given, one is able to combine them by a geodesic in S. Notice that the image of each such geodesic always contains in the set B. Now, a given geodesic triangle Δ(e 1 , e 2 , e 3 ) in a geodesic metric space S is composed of three elements e 1 , e 2 , e 3 in S and a choice of three geodesic segments [e 1 , e 2 ], [e 2 , e 3 ], [e 3 , e 1 ] combining them.
A triangle Δ(e 1 , e 2 , e 3 ) in the Euclidean plane R 2 is called a comparison triangle for a given geodesic triangle Δ(e 1 , e 2 , e 3 ) provided that An element c ∈ [e 1 , e 2 ] is known as a comparison element for c ∈ [e 1 , e 2 ] provided that d(e 1 , c) � d R 2 (e 1 , c). Comparison elements on [e 2 , e 3 ] and [e 3 , e 1 ] can be stated in the similar way. Definition 1. Suppose Δ(e 1 , e 2 , e 3 ) is a given geodesic triangle in a metric space (S, d). We say that Δ(e 1 , e 2 , e 3 ) is endowed with the CAT(0) inequality whenever for e, e ′ ∈ Δ(e 1 , e 2 , e 3 ) and for their comparison elements, namely, e, e ′ ∈ Δ(e 1 , e 2 , e 3 ), it follows that Notice 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 [24]. 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 [24][25][26][27].
We now state a result from [28].
Assume that ∅ ≠ B ⊆ S is convex and closed in a CAT(0) space and b k ⊆ S is bounded. Fix b ∈ S, and set We normally denote and define the asymptotic radius of the sequence b k wrt B as We normally denote and define the asymptotic center of the sequence b k wrt B as It is known that the asymptotic center is simply denoted by A( b k ) wrt S. Interestingly, the set A(B, b k ) contains unique elements in the setting of complete CAT(0) spaces (see [29]).

Definition 2.
A bounded sequence b k of elements of a CAT(0) space S is called Δ-convergent to q 0 ∈ S provided that q 0 is the unique asymptotic center for every subse- Every CAT(0) space S enjoys the Opial-like property [30], that is, if b k ⊆S is Δ-convergent to q 0 , then for every choice of p 0 ≠ q 0 ∈ S, it follows that Lemma 2 (see [31]). Suppose S is a CAT(0) space and b k ⊆S is bounded. en, b k always has a Δ-convergent subsequence.
Lemma 3 (see [32]). Suppose S is a CAT(0) space, B⊆S is closed and convex, and b k ⊆B is bounded. en, the asymptotic center of the sequence b k is always in B.
e following characterizations of Garcia-Falset maps can be found in [12]. (i) If G is a Suzuki mapping, then G is also a Garcia-Falset map (ii) If G is a Garcia-Falset map having F G ≠ ∅, then for every choice of b ∈ B and q 0 ∈ F G , one has d(Gb, Gq 0 ) ≤ d(b, q 0 ) (iii) If G is a Garcia-Falset map, then the set F G is closed We know the following characterization of the CAT(0) space from [33].

Lemma 4.
Assume that (S, d) is a given complete CAT(0) space, and fix b ∈ S. Suppose 0 < y ≤ η k ≤ z < 1 and r k , s k are two sequences of elements of S such that

Convergence Theorems in CAT(0) Spaces
e aim of this section is to establish Δ and strong convergence of the Sahu-akur iterative scheme for Garcia-Falset maps. roughout the section, we shall denote simply by S a complete CAT(0) space. Notice that Sahu-akur iterative process (3) in the framework of CAT(0) spaces is written as follows: where α k , β k , c k ∈ (0, 1). We first establish a key lemma as follows.
Lemma 5. Suppose B be any nonempty closed convex subset of S, and assume that G: B ⟶ B is a Garcia-Falset map having F G ≠ ∅. If the sequence b k is defined by equation (12), then lim k⟶∞ d(b k , q 0 ) exists for each choice of q 0 ∈ F G .
Proof. Suppose q 0 ∈ F G . Keeping Proposition 1 (ii) in mind, one has Using the above inequalities, we have Mathematical Problems in Engineering Hence, the real sequence d(b k , q 0 ) is nonincreasing and bounded.
us, we conclude that lim k⟶∞ d(b k , q 0 ) exists for each choice of q 0 ∈ F G .
Using Lemma 5 and the definition of the Garcia-Falset map, we establish the following result in CAT(0) spaces. Proof. Suppose that the sequence b k is bounded and lim k⟶∞ d(b k , Gb k ) � 0. Fix an element q 0 ∈ A(B, b k ). We are going to show that q 0 � Gq 0 . Since G is a Garcia-Falset map, Hence, it is seen that Gq 0 ∈ A(B, b k ), but A(B, b k ) is singleton, and one can conclude that q 0 � Gq 0 . us, q 0 is the element of F G .
Conversely, let the set F G ≠ ∅. We are going to show that b k is bounded and lim k⟶∞ d(b k , Gb k ) � 0. Fix q 0 ∈ F G ; then, by Lemma 5, lim k⟶∞ d(b k , q 0 ) exists, and b k is bounded. Put e proof of Lemma 5 suggests that By using Proposition 1 (ii), one has Also, the proof of Lemma 5 suggests that From equations (17) and (19), we obtain From equation (20), we have Applying Lemma 4, we obtain is completes the proof. We now establish a strong convergence for Garcia-Falset mappings on compact domains in CAT(0) spaces.

Theorem 2.
Assume that B is a nonempty convex compact subset of S. Suppose G and b k are as given in eorem 1 and F G ≠ ∅. en, b k converges strongly to a point of F G .
Proof. By the compactness property of the set B, we can construct a subsequence b k l of b k such that lim l⟶∞ d(b k l , p 0 ) � 0, for some p 0 ∈ B. Since the self-map G is a Garcia-Falset map, there exists a real constant μ ≥ 1, with In the view of eorem 1, lim l⟶∞ d(b k l , Gb k l ) � 0. Now, using lim l⟶∞ d(b k l , Gb k l ) � 0 and lim l⟶∞ d(b k l , p 0 ) � 0, we have from equation (23) lim l⟶∞ d(b k l , Gp 0 ) � 0, but S is a metric space, so the limit of b k l is unique, that is, Gp 0 � p 0 . Hence, p 0 ∈ F G . By Lemma 5, lim k⟶∞ d(b k , p 0 ) exists. Hence, p 0 is the strong limit of b k . 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 [34].

Definition 3.
Assume that B is a nonempty subset of S. One says that a self-map G of B has condition I whenever there is a nondecreasing map μ which satisfies μ(p) � 0 if and only if p � 0 and μ(p) > 0 for every choice of p > 0 and d(b, Gb) ≥ μ (d(b, F G )) for each choice of b ∈ B. Now, we state and prove a strong convergence result based on condition I.
Condition I of G suggests Using equation (24) and condition I, one has However, the map μ: [0, ∞) ⟶ [0, ∞) is nondecreasing with μ(0) � 0 and μ(p) > 0 for each choice of p > 0, so e conclusions of eorem 3 suggest that b k is strongly convergent in the set F G .
Finally, we suggest Δ-convergence for maps having (E)condition.

Theorem 5.
Assume that B is a nonempty closed convex subset of S. Suppose G and b k are as given in eorem 1 and F G ≠ ∅. en, b k Δ converges to a point of F G .
Proof. By eorem 1, one can conclude that b k is bounded and is such that lim k⟶∞ d(b k , Gb k ) � 0. We may suppose that ω Δ ( b k ) � ⋃ A( s k ), for every subsequence s k of b k . e next purpose is to obtain ω Δ ( b k )⊆F G . If we select any s ∈ ω Δ ( b k ), then there exists a subsequence, namely, s k , of b k with the property A( s k ) � s { }. Using Lemmas 2 and 3, we may conclude that a subsequence r k of s k exists having some Δ-limit r in B, but eorem 1 suggests that lim k⟶∞ d(r k , Gr k ) � 0, and also, G is endowed with the (E)-condition, so we have d r k , Gr ≤ μ d r k , Gr k + d r k , r .
By considering lim sup on both sides of equation (28), we obtain r ∈ F G . Lemma 5 suggests that lim k⟶∞ d(r k , r) exists. We need to show that s � r. Assume, on the contrary, that s is different from r. So, by uniqueness of asymptotic centers, one has Consequently, limsup k⟶∞ d(r k , r) < limsup k⟶∞ d(r k , r).
is is a contradiction, and so, one can conclude that s � r ∈ F G and ω Δ ( b k )⊆F G .
It is now remaining to prove b k is a Δ-convergent sequence in the set F G . For this purpose, we need ω Δ ( b k ) which is a singleton set. If s k is any subsequence of b k , then by Lemmas 2 and 3, we can choose the Δ-convergent subsequence r k of s k having some Δ − lim r in B. Assume that A( s k ) � s { } and A( b k ) � q . We have already showed that s � r and r ∈ F G . We are going to prove that q � r. Suppose not, then lim k⟶∞ d(b k , r) exists, and also, the asymptotic centers are singleton, and we have which is a contradiction.
us, q � r ∈ F G . Hence, is completes the proof.

Numerical Example
We now furnish a new numerical example of Garcia-Falset maps and show that its Sahu-akur iterative scheme is more effective than the other well-known existing iterative schemes.
To observe that the map G is a Garcia-Falset map, we will establish d(b, Gb ′ ) ≤ μ d(b, Gb) + d(b, b ′ ) for every choice of b, b ′ ∈ B and some μ ≥ 1. We choose μ � 2 and divide the proof in parts as follows: Mathematical Problems in Engineering (P 4 ) Let b ∈ B 2 and b ′ ∈ B 1 , so Gb ′ � ((b ′ + 4)/2) and Gb � 4. en, e cases discussed above show that G is a Garcia-Falset map on the set B. By choosing b � 8.2 and b ′ � 10, we may   d(b, b ′ ), that is, G is not a Suzuki map on B. We now choose α k � 0.85 and β k � 0.65 and c k � 0.45 for every natural number k. In Table 1, strong convergence of Sahu-akur iteration (ST) [20,21] is compared with the other existing iterative schemes such as Picard [35], Mann [14], Ishikawa [15], Noor [17], Agarwal et al. et al. [16], Abbas and Nazir [18], and akur et al. [19]. Both Table 1 and Figure 1 well suggest the effectiveness of Sahu-akur iteration in the class of Garcia-Falset maps.

Conclusion
e 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 [36], 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 [23], Usurelu et al. established some convergence results for Garcia-Falset maps under Sahu-akur 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-akur 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. us, our results simultaneously improve and extend the corresponding results in [7-9, 11, 18-22, 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 [38].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
All the authors contributed equally to this article.