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

Authors: Kifayat Ullah; Junaid Ahmad; Muhammad Arshad; Manuel De la Sen
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this article, we suggest someΔ 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.


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## Body

## 1. Introduction

LetS,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 dGb,Gb′≤rdb,b′. The self-map G is regarded as nonexpansive if and only if dGb,Gb′≤db,b′. A point, namely, q0∈B, is called a fixed point for the self-map G of B if and only if Gq0=q0, and the notation FG 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, q0, and for every choice of initial value, Picard iterative method (sometimes called the successive approximation method) converges to this q0. In 1930, Caccioppoli [2] 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 [3] and Göhde [4] in the year 1965. They 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(1)12db,Gb≤db,b′⇒dGb,Gb′≤db,b′.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. [8–11] 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(2)db,Gb′≤μdb,Gb+b,b′,for someμ≥1.Since by Suzuki [6], every Suzuki map of a nonempty subset B of a metric space satisfies db,Gb′≤3db,Gb+b,b′ for every two elements b,b′∈B. 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. Thus, 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.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 [13]. 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 [14], Ishikawa [15], Agarwal et al. [16], Noor [17], Abbas and Nazir [18], and Thakur et al. [19]). Among these iterative processes, Sahu et al. [20] and Thakur et al. [21] independently suggested an effective three-step iterative scheme for approximating fixed points of nonexpansive maps:(3)b1=b∈B,gk=1−γkbk+γkGbk,sk=1−βkgk+βkGgk,bk+1=1−αkGgk+αkGsk,k∈ℙ,where αk,βk,γk∈0,1 and ℙ denotes the set of all positive integers.In [21], 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 [22] improved and extended the main outcome of Thakur 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. The 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 Thakur 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–Thakur iterative process (3) is more effective than the many other iterative schemes.


## 2. Preliminaries

Take a closed interval0,v in the set ℝ, and suppose b,b′ are any elements of a metric space S=S,d. Then, a map g:0,v⟶S is called geodesic provided that g0=b, gv=b′, and dgr1,gr2=r1−r2 for every choice of two points r1,r2∈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 triangleΔe1,e2,e3 in a geodesic metric space S is composed of three elements e1,e2,e3 in S and a choice of three geodesic segments e1,e2,e2,e3,e3,e1 combining them.A triangleΔ¯e¯1,e¯2,e¯3 in the Euclidean plane ℝ2 is called a comparison triangle for a given geodesic triangle Δe1,e2,e3 provided that(4)dℝ2e¯1,e¯2=de2,e3,dℝ2e¯2,e¯3=de2,e3,dℝ2e¯3,e¯1=de3,e1.An elementc¯∈e¯1,e¯2 is known as a comparison element for c∈e1,e2 provided that de1,c=dℝ2e¯1,c¯. Comparison elements on e¯2,e¯3 and e¯3,e¯1 can be stated in the similar way.Definition 1.
SupposeΔe1,e2,e3 is a given geodesic triangle in a metric space S,d. We say that Δe1,e2,e3 is endowed with the CAT(0) inequality whenever for e,e′∈Δe1,e2,e3 and for their comparison elements, namely, e¯,e′¯∈Δ¯e¯1,e¯2,e¯3, it follows that(5)de,e′=dℝ2e¯,e¯′.
Notice that a geodesic spaceS 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 [24–27].
We now state a result from [28].Lemma 1.
SupposeS,d is a given CAT(0) space.(i)
Ifb,b′∈S and κ∈0,1 is fixed, then there exists a unique element, namely, g∈b,b′, such that(6)db,g=κdb,b′,db′,g=1−κdb,b′.We shall normally represent by1−κb⊕κb′ the unique element g satisfying (6).(ii)
Ifb,b′,a∈S and κ∈0,1 is fixed, then(7)da,κb⊕1−κb′≤κda,b+1−κb′.Assume that∅≠B⊆S is convex and closed in a CAT(0) space and bk⊆S is bounded. Fix b∈S, and set(8)rb,bk=limsupk⟶∞dbk,b.We normally denote and define the asymptotic radius of the sequencebk wrt B as(9)rB,bk=infrb,bk:b∈B.We normally denote and define the asymptotic center of the sequencebk wrt B as(10)AB,bk=b∈B:rb,bk=rB,bk.It is known that the asymptotic center is simply denoted byAbk wrt S. Interestingly, the set AB,bk contains unique elements in the setting of complete CAT(0) spaces (see [29]).Definition 2.
A bounded sequencebk of elements of a CAT(0) space S is called Δ-convergent to q0∈S provided that q0 is the unique asymptotic center for every subsequence sk of bk. When q0 is Δ−lim of bk, then we write Δ−limkbk=q0.
Every CAT(0) spaceS enjoys the Opial-like property [30], that is, if bk⊆S is Δ-convergent to q0, then for every choice of p0≠q0∈S, it follows that(11)limsupk⟶∞dbk,q0<limsupk⟶∞dbk,p0.Lemma 2.
(see [31]). Suppose S is a CAT(0) space and bk⊆S is bounded. Then, bkalways has a Δ-convergent subsequence.Lemma 3.
(see [32]). Suppose S is a CAT(0) space, B⊆S is closed and convex, and bk⊆B is bounded. Then, the asymptotic center of the sequence bk is always in B.The following characterizations of Garcia-Falset maps can be found in [12].Proposition 1.
LetB be a nonempty subset of a complete CAT(0) space and G:B⟶B.(i)
IfG is a Suzuki mapping, then G is also a Garcia-Falset map(ii)
IfG is a Garcia-Falset map having FG≠∅, then for every choice of b∈B and q0∈FG, one has dGb,Gq0≤db,q0(iii)
IfG is a Garcia-Falset map, then the set FG is closed
We know the following characterization of the CAT(0) space from [33].Lemma 4.
Assume thatS,d is a given complete CAT(0) space, and fix b∈S. Suppose 0<y≤ηk≤z<1 and rk, sk are two sequences of elements of S such that limsupk⟶∞drk,b≤ξ, limsupk⟶∞dsk,b≤ξ, and limk⟶∞dηkrk⊕1−ηksk,b=ξ for some real number ξ≥0; then, limk⟶∞drk,sk=0.


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

The aim of this section is to establishΔ 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:(12)b1=b∈B,gk=1−γkbk⊕γkGbk,sk=1−βkgk⊕βkGgk,bk+1=1−αkGgk⊕αkGsk,k∈ℙ,where αk,βk,γk∈0,1.We first establish a key lemma as follows.Lemma 5.
SupposeB be any nonempty closed convex subset of S, and assume that G:B⟶B is a Garcia-Falset map having FG≠∅. If the sequence bk is defined by equation (12), then limk⟶∞dbk,q0 exists for each choice of q0∈FG.Proof.
Supposeq0∈FG. Keeping Proposition 1 (ii) in mind, one has(13)dgk,q0=d1−γkbk⊕γkGbk,q0≤1−γkdbk,q0+γkdGbk,q0≤1−γkdbk,q0+γkdbk,q0≤dbk,q0,dsk,q0=d1−βkgk⊕βkGgk,q0≤1−βkdgk,q0+βkdGgk,q0≤1−βkdgk,q0+βkdgk,q0≤dgk,q0.
Using the above inequalities, we have(14)dbk+1,q0=d1−akGgk⊕αkGsk,q0≤1−αkdGgk,q0+αkdGsk,q0≤1−αkdgk,q0+αkdsk,q0≤1−αkdgk,q0+αkdgk,q0=dgk,q0≤dbk,q0.
Hence, the real sequencedbk,q0 is nonincreasing and bounded. Thus, we conclude that limk⟶∞dbk,q0 exists for each choice of q0∈FG.
Using Lemma5 and the definition of the Garcia-Falset map, we establish the following result in CAT(0) spaces.Theorem 1.
LetB be a nonempty convex closed subset of S, and let G:B⟶B be a Garcia-Falset operator. For arbitrary chosen b1∈B, let the sequence bk be defined by equation (12). Then, FG≠∅ if and only if bk is bounded and limk⟶∞dbk,Gbk=0.Proof.
Suppose that the sequencebk is bounded and limk⟶∞dbk,Gbk=0. Fix an element q0∈AB,bk. We are going to show that q0=Gq0. Since G is a Garcia-Falset map,(15)rGq0,bk=limsupk⟶∞dbk,Gq0≤μlimsupk⟶∞dbk,Gbk+limsupk⟶∞dbk,q0=limsupk⟶∞dbk,q0=rq0,bk.
Hence, it is seen thatGq0∈AB,bk, but AB,bk is singleton, and one can conclude that q0=Gq0. Thus, q0 is the element of FG.
Conversely, let the setFG≠∅. We are going to show that bk is bounded and limk⟶∞dbk,Gbk=0. Fix q0∈FG; then, by Lemma 5, limk⟶∞dbk,q0 exists, and bk is bounded. Put(16)limk⟶∞dbk,q0=ξ.
The proof of Lemma5 suggests that(17)dgk,q0≤dbk,q0⇒limsupk⟶∞dgk,q0≤limsupk⟶∞dbk,q0=ξ.
By using Proposition1 (ii), one has(18)limsupk⟶∞dGbk,q0≤limsupk⟶∞dbk,q0=ξ.
Also, the proof of Lemma5 suggests that(19)dbk+1,q0≤dgk,q0⇒ξ≤liminfk⟶∞dgk,q0.
From equations (17) and (19), we obtain(20)ξ=limk⟶∞dgk,q0.
From equation (20), we have(21)ξ=limk⟶∞dgk,q0=limk⟶∞d1−γkbk⊕γkGbk,q0.
Applying Lemma4, we obtain(22)limk⟶∞dbk,Gbk=0.
This completes the proof.
We now establish a strong convergence for Garcia-Falset mappings on compact domains in CAT(0) spaces.Theorem 2.
Assume thatB is a nonempty convex compact subset of S. Suppose G and bk are as given in Theorem 1 and FG≠∅. Then, bk converges strongly to a point of FG.Proof.
By the compactness property of the setB, we can construct a subsequence bkl of bk such that liml⟶∞dbkl,p0=0, for some p0∈B. Since the self-map G is a Garcia-Falset map, there exists a real constant μ≥1, with(23)dbkl,Gp0≤μdbkl,Gbkl+dbkl,p0.
In the view of Theorem1, liml⟶∞dbkl,Gbkl=0. Now, using liml⟶∞dbkl,Gbkl=0 and liml⟶∞dbkl,p0=0, we have from equation (23) liml⟶∞dbkl,Gp0=0, but S is a metric space, so the limit of bkl is unique, that is, Gp0=p0. Hence, p0∈FG. By Lemma 5, limk⟶∞dbk,p0 exists. Hence, p0 is the strong limit of bk.Theorem 3.
Assume thatB is a nonempty closed convex subset of S. Suppose G and bk are as given in Theorem 1. If FG≠∅ and liminfk⟶∞dbk,FG=0, then bk converges strongly to a point of FG.Proof.
Since the proof is elementary, we skip it.
We now give the complete definition of conditionI, which was originally introduced by Sentor and Dotson in [34].Definition 3.
Assume thatB 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 μp=0 if and only if p=0 and μp>0 for every choice of p>0 and db,Gb≥μdb,FG for each choice of b∈B.
Now, we state and prove a strong convergence result based on conditionI.Theorem 4.
Assume thatB is a nonempty closed convex subset of S. Suppose G and bk are as given in Theorem 1 and FG≠∅. Then, bk converges strongly to a point of FG if G has condition I.Proof.
The conclusions of Theorem1 provide us(24)liminfk⟶∞dbk,Gbk=0.
ConditionI of G suggests(25)dbk,Gbk≥μdbk,FG.
Using equation (24) and condition I, one has(26)liminfk⟶∞μdbk,FG=0.
However, the mapμ:0,∞⟶0,∞ is nondecreasing with μ0=0 and μp>0 for each choice of p>0, so(27)liminfk⟶∞dbk,FG=0.
The conclusions of Theorem3 suggest that bk is strongly convergent in the set FG.
Finally, we suggestΔ-convergence for maps having (E)-condition.Theorem 5.
Assume thatB is a nonempty closed convex subset of S. Suppose G and bk are as given in Theorem 1 and FG≠∅. Then, bkΔ converges to a point of FG.Proof.
By Theorem1, one can conclude that bk is bounded and is such that limk⟶∞dbk,Gbk=0. We may suppose that ωΔbk=∪Ask, for every subsequence sk of bk. The next purpose is to obtain ωΔbk⊆FG. If we select any s∈ωΔbk, then there exists a subsequence, namely, sk, of bk with the property Ask=s. Using Lemmas 2 and 3, we may conclude that a subsequence rk of sk exists having some Δ-limit r in B, but Theorem 1 suggests that limk⟶∞drk,Grk=0, and also, G is endowed with the (E)-condition, so we have(28)drk,Gr≤μdrk,Grk+drk,r.
By considering lim sup on both sides of equation (28), we obtain r∈FG. Lemma 5 suggests that limk⟶∞drk,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(29)limsupk⟶∞drk,r<limsupk⟶∞drk,s≤limsupk⟶∞dsk,s<limsupk⟶∞dsk,r=limsupk⟶∞dbk⋅r=limsupk⟶∞drk,r.
Consequently,limsupk⟶∞drk,r<limsupk⟶∞drk,r. This is a contradiction, and so, one can conclude that s=r∈FG and ωΔbk⊆FG.
It is now remaining to provebk is a Δ-convergent sequence in the set FG. For this purpose, we need ωΔbk which is a singleton set. If sk is any subsequence of bk, then by Lemmas 2 and 3, we can choose the Δ-convergent subsequence rk of sk having some Δ−limr in B. Assume that Ask=s and Abk=q. We have already showed that s=r and r∈FG. We are going to prove that q=r. Suppose not, then limk⟶∞dbk,r exists, and also, the asymptotic centers are singleton, and we have(30)limsupk⟶∞drk,r<limsupk⟶∞drk,q≤limsupk⟶∞dbk,q<limsupk⟶∞dbk,r=limsupk⟶∞drk⋅r,which is a contradiction. Thus, q=r∈FG. Hence, ωΔb=q. 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-mapG of B=4,10 by the rule Gb=b+4/2 for b∈B1=4,10 and Gb=4 for b∈B2=10. To observe that the map G is a Garcia-Falset map, we will establish db,Gb′≤μdb,Gb+db,b′ for every choice of b,b′∈B and some μ≥1. We choose μ=2 and divide the proof in parts as follows:P1 Let b,b′∈B2; then, Gb=4=Gb′. Then,(31)db,Gb′=b−Gb′=b−4=b−Gb≤2b−Gb≤2b−Gb+b−b′=2db,Gb+db,b′.P2 Let b,b′∈B1, so Gb=b+4/2 and Gb′=b′+4/2. Then,(32)db,Gb′=b−Gb′≤b−Gb+Gb−Gb′=b−Gb+b+42−b′+42=b−Gb+12b−b′≤b−Gb+b−b′≤2b−Gb+b−b′.P3 Let b∈B1 and b′∈B2, so Gb=b+4/2 and Gb′=4. Then,(33)db,Gb′=b−Gb′=b−4=2b−42=2b−b+42=2b−Gb≤2b−Gb+b−b′=2db,Gb+db,b′.P4 Let b∈B2 and b′∈B1, so Gb′=b′+4/2 and Gb=4. Then,(34)db,Gb′=b−Gb′=b−b′+42=b+b−b′−42≤b−42+b−b′2=12b−Gb+12b−b′≤2b−Gb+b−b′=2db,Gb+db,b′.
The cases discussed above show thatG is a Garcia-Falset map on the set B. By choosing b=8.2 and b′=10, we may observe that 1/2db,Gb≤db,b′ and dGb,Gb′>db,b′, that is, G is not a Suzuki map on B. We now choose αk=0.85 and βk=0.65 and γk=0.45 for every natural number k. In Table 1, strong convergence of Sahu–Thakur 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 Thakur et al. [19]. Both Table 1 and Figure 1 well suggest the effectiveness of Sahu–Thakur iteration in the class of Garcia-Falset maps.Table 1
Some values generated by Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur, and Sahu-Thakur (ST) iterative schemes toq0=4 of the mapping G as given in Example 1.
kPicardMannIshikawaNoorAgarwalAbbasThakurST14.54.54.54.54.54.54.54.524.25004.28754.21844.20294.18094.18074.14874.140234.12504.16534.09544.08234.06554.06534.04424.039344.06254.09504.04164.03344.02364.02364.01314.011054.03134.05464.01824.01364.00854.00854.00394.003164.01564.03144.00794.00554.00314.00314.00124.000974.00784.01804.00344.00224.00114.00114.00034.000284.00394.01034.00154.00094.00044.00044.00014.000094.00204.00594.00064.00034.00014.00014.00004.0000104.00104.00344.00024.00014.00004.00004.00004.0000114.00054.00194.00014.00004.00004.00004.00004.0000124.00024.00114.00004.00004.00004.00004.00004.0000134.00014.00064.00004.00004.00004.00004.00004.0000144.00004.00044.00004.00004.00004.00004.00004.0000154.00004.00024.00004.00004.00004.00004.00004.0000164.00004.00014.00004.00004.00004.00004.00004.0000174.00004.00004.00004.00004.00004.00004.00004.0000Figure 1
Convergence behaviors of different iterative schemes to a fixed pointq0=4 of the self-map G.


## 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 [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–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 [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].



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