# Iterative Construction of Fixed Points for Operators Endowed with ConditionE in Metric Spaces

Authors: Junaid Ahmad; Kifayat Ullah; Hüseyin Işik; Muhammad Arshad; Manuel de la Sen
Journal: Advances in Mathematical Physics
 (2021)

---

## Abstract

We consider the class of mappings endowed with the conditionE in a nonlinear domain called 2-uniformly convex hyperbolic space. We provide some strong and Δ-convergence theorems for this class of mappings under the Agarwal iterative process. In order to support the main outcome, we procure an example of mappings endowed with the condition E and prove that its Agarwal iterative process is more effective than Mann and Ishikawa iterative processes. Simultaneously, our results hold in uniformly convex Banach, CAT(0), and some CAT(κ) spaces. This approach essentially provides a new setting for researchers who are working on the iterative procedures in fixed point theory and applications.


---

## Body

## 1. Introduction

Choose a metric spaceZ and ∅≠S⊆Z. Then, a self-map L:S⟶S is called nonexpansive if
(1)qLs,Ls′≤qs,s′,foralls,s′∈S.As many know, Browder [1] (see also Kirk [2] and Gohde [3]) achieved an elementary fixed point theorem for nonexpansive map L:S⟶S in the case when S is closed bounded convex in any given uniformly convex Banach space (UCBS). This remarkable theorem began the research of different results for nonexpansive maps. Among other things, Kirk [4] obtained this theorem in the nonlinear setting of CAT(0) metric spaces. The class of nonexpansive mappings is very important because of its useful applications in many fields of applied sciences. Thus, it is very natural and essential to study the extension of these maps. In 2008, Suzuki [5] considered an extension of nonexpansive maps by weakening the inequality of the nonexpansive maps in the following way: a self-map L:S⟶S is said to hold condition C if any given s,s′∈S,
(2)12qs,Ls≤qs,s′⇒qLs,Ls′≤qs,s′.Obviously, a nonexpansive mapL has condition (C), but an example in [5] shows the inverse is not valid in general. In [6], Nanjaras et al. improved the main outcome of Suzuki [5] to the nonlinear setting of CAT(0) spaces. In [7], Phuengrattana used Ishikawa iteration [8] for reckoning of fixed points for the class of Suzuki mappings in a Banach space. Also, Basarir and Sahin [9] used Agarwal et al.’s [8] iteration for reckoning fixed points of Suzuki mappings in the nonlinear setting of CAT(0) spaces.In [10], Garcia-Falset et al. suggested a new type of maps which is properly more general than Suzuki maps. A self-map L of S is called Garcia-Falset nonexpansive or said to be endowed with the condition E if any given s,s′∈S, there is a μ≥1 such that
(3)qs,Ls′≤μqs,Ls+qs,s′.Remark 1.
IfL is nonexpansive, then it is easy to show qs,Ls′≤qs,Ls+qs,s′ for each s,s′∈S. Hence, every nonexpansive map L has condition E with μ=1. Similarly, if L is Suzuki map, then, it satisfies qs,Ls′≤3qs,Ls+qs,s′ for every s,s′∈S. Hence, every Suzuki map L has condition E with μ=3. Interestingly, the example constructed in this paper shows that there exists maps satisfying condition E but not the converse.After establishing the existence of a fixed point result for an operator, a natural question arises on how to compute it by any appropriate numerical method. To work on such type of problems is not easy. The well-known Banach Contraction Theorem (BCT) essentially suggests the Picard iterationyk+1=Lyk to compute the unique fixed point of a given contraction in a metric space. Bagherboum [11] used Ishikawa iteration [12] for computing fixed points of Garcia-Falset mappings in the nonlinear setting of Busemann spaces. We know that the Picard iteration fails for solving nonexpansive problems. Now, we discuss some other methods which are different from the Picard iteration process.One of the elementary iterative processes is due to Mann [13] stated as follows:
(4)y1∈S,yk+1=1−αkyk⊕αkLyk,where αk∈0,1.The Mann iteration (4) was extended to the setting of two steps by Ishikawa [12] as follows:
(5)y1∈S,yk+1=1−αkyk⊕αkLwk,wk=1−βkyk⊕βkLyk,where αk,βk∈0,1.In 2007, Agarwal et al. [8] studied the following iteration for contractions and observed that it is better than the above iterative processes:
(6)y1∈S,yk+1=1−αkLyk⊕αkLwk,wk=1−βkyk⊕βkLyk,whereαk,βk∈0,1.In this research, we consider the Agarwal iterative process (6) for the larger class of maps due to Garcia-Falset et al. [10]. We use a very general ground space called 2-uniformly convex hyperbolic metric space for establishing the main outcome. To support our results, we provide an example of maps with condition E and prove that its Agarwal iterative process (6) is more effective than Mann (4) and Ishikawa iterative (5) processes. Simultaneously, our results hold in uniformly convex Banach, CAT(0), and some CAT(κ) spaces. In this way, we extend the corresponding results of Bagherboum [11], Phuengrattana [7], Basarir and Sahin [9], and Nanjaras et al. [6] as we consider the general setting of domain, larger class of maps, and faster iteration process.Now, we present some well-known definitions and results which will be either used in the main results or to understand the given concept herein.Definition 2 (see [14]).
A given metric spaceZ,q is called Kolenbach–hyperbolic space or simply hyperbolic space when there is a map C:Z×Z×0,1⟶Z such that for all s,s′,u,p∈Z and i,j∈0,1, we have
(C1) qu,Cs,s′,i≤1−iqu,s+iqu,s′;
(C2) qCs,s′,i,Cs,s′,j=i−jqs,s′;
(C3) Cs,s′,i=Cs′,s,1−i;
(C4) qCs,u,i,Cs′,p,i≤1−iqs,s′+iqu,p.
Notice that, for givens,s′∈Z and i∈0,1, we shall use 1−is⊕is′ for Cs,s′,i. Also, the property (C1) gives
(7)qs,1−is⊕is′=iqs,s′andqs′,1−is⊕is′=1−iqs,s′.Keep in mind that∅≠S⊆Z is said to be convex in Z if for every s,s′∈S, s,s′=1−σs⊕σs′:σ∈0,1⊆S.Definition 3.
A given hyperbolic spaceZ,q,C is said to be a uniformly convex hyperbolic space (UCHS), if there exist r>0, σ∈0,1 and ε∈0,2 such that for each s,s′,p∈Z satisfying qs,p≤r, qs′,p≤r and qs,s′≥2ε follows that:
(8)q12s⊕12s′,p≤1−σr.Moreover, iff:0,∞×0,2⟶0,1 is given such that σ=fr,ε for all r∈0,∞ and ε∈0,2, then, f is called a modulus of uniform convexity (MUC). Also, f is called monotone modulus of uniform convexity (MMUC) iff nonincreasing in the variable r for every chosen fixed ε.Definition 4.
Suppose a pairZ,q is a given hyperbolic space. For any r∈0,∞ and ε∈0,2, define:
(9)φr,ε=inf12q2s,p+12q2s′,p−q212s⊕12s′,p,and keeping in mind that infimum is taken over every s,s′,p∈Z satisfying qs,p≤r, qs′,p≤r, and qs,s′≥rε. Then, Z,q is called 2-uniformly convex hyperbolic space (2-UCHS) provided that
(10)cM=infφr,εr2ε2:r∈0,∞,ε∈0,2>0.The following remark suggests that 2-UCHS is a more general domain than many other linear and nonlinear spaces.Remark 5.
Any given UCBS, CAT(0) space, and CAT(κ) spaces having κ>0 and diamZ≤π/2−ε/κ1/2, ε∈0,π/2 can be considered as a 2-UCHS (see [15–17]).Definition 6.
Choose a complete 2-UCHSZ such that yk⊆Z is bounded and ∅≠S⊆Z is closed as well as convex. The asymptotic radius of yk corresponding to S is rS,yk=inflimsupk⟶∞qyk,u:u∈S. Moreover, the asymptotic center of yk corresponding to S is the set AS,yk=u∈S:limsupk⟶∞qyk,u=rS,yk. In addition, if the underlying space has MMUC, then the set AS,yk is singleton (see, e.g., [18]).Definition 7.
Choose a complete 2-UCHSZ such that ∅≠S⊆Z is closed as well as convex. Fix an element w∈S. Assume that yk is a bounded sequence in Z. Then, yk is called Δ-convergent to wiff the asymptotic center AS,zk=w for all subsequence zk of yk. When w is Δ-limit of yk, then we shall write Δ−limk⟶∞yk=w.Lemma 8 (see [19]).
Any bounded sequence in a given 2-UCHS having MMUC has aΔ-convergent subsequence.Lemma 9 (see [20]).
Choose a complete 2-UCHSZ endowed with MMUC such that ∅≠S⊆Z and yk⊆S is bounded. Then, the asymptotic center of yk is always in S.
Mappings endowed with conditionE enjoy the following useful properties which we have combined in a proposition form. For details, we refer the reader to [10].Proposition 10.
Choose a complete 2-UCHSZ such that ∅≠S⊆Z. Suppose L:S⟶S.
(i)
When the mapL has condition E with nonempty fixed point set, then for each s∈S and ω∈FL, we have qLs,Lω≤qs,ω(ii)
When the mapL has condition E then fixed point set of S is always closed in SWe now state an important characterization of a 2-UCHS, which was proved in [15].Lemma 11.
Choose a 2-UCHSZ,q. Then, the following inequality holds
(11)q21−ξs⊕ξs′,g≤1−ξq2s,g+ξq2s′,g−4cMξ1−ξq2s,s′,for each ξ∈0,1 and s,s′,g∈Z.


## 2. Approximation Results

We now want to show our desirable strong andΔ-convergence results by considering the Agarwal iteration (6). It should be noted that Z will be considered complete and endowed with the MMUC. The notation FL will simply denote the set of all fixed points of L.We suggest the following useful lemma which will provide a key role in main outcome.Lemma 12.
Choose a 2-UCHSZ such that ∅≠S⊆Z is closed as well as convex. Furthermore, let L:S⟶S be endowed with condition E having FL≠∅. If yk is the Agarwal iterates (6), then, limk⟶∞qyk,ω exists for all ω∈FL.Proof.
Assume thatω∈FL. By using Proposition 10 (i), one has
(12)qwk,ω=q1−βkyk⊕βkLyk,ω≤1−βkqyk,ω+βkqLyk,ω≤1−βkqyk,ω+βkqyk,ω≤qyk,ω.
Hence,(13)qyk+1,ω=q1−αkLyk⊕αkLwk,ω≤1−αkqLyk,ω+αkqLwk,ω≤1−αkqyk,ω+αkqwk,ω≤qyk,ω.
From the above observations, we haveqyk+1,ω≤qyk,ω. Hence, the real sequence qyk,ω is bounded below by 0 and nonincreasing as well, so limk⟶∞qyk,ω exists for every ω∈FL.The existence of a fixed point for a self-mapL:S⟶S having condition E in the nonlinear domain 2-UCBS can be established by the following way. This result provides a fundamental key for establishing the main results in the sequel.Theorem 13.
Choose a 2-UCHSZ such that ∅≠S⊆Z is closed as well as convex. Furthermore, let L:S⟶S be endowed with condition E. If yk is the Agarwal iterates (6) with restriction αk,βk∈a,b⊂0,1. Then, FL≠∅iffyk is bounded and limk⟶∞qyk,Lyk=0.Proof.
Suppose that the iterative sequenceyk is bounded having limk⟶∞qLyk,yk=0. We may fix an element, namely, ω∈AS,yk. By using condition E of L, we have
(14)rLω,yk=limsupk⟶∞qyk,Lω≤μlimsupk⟶∞qLyk,yk+limsupk⟶∞qyk,ω=limsupk⟶∞qyk,ω=rω,yk.
It follows thatLω∈AS,yk. Since AS,yk is singleton set, one has Lω=ω. Thus, FL≠∅. Conversely, we suppose that FL≠∅ and ω∈FL. Then, by the proof of Lemma 12, yk is bounded. By Lemma 11, we have
(15)q2wk,ω≤1−βkq2yk,ω+βkq2Lyk,ω−4cMβk1−βkq2yk,Lyk≤1−βkq2yk,ω+βkq2yk,ω−4cMβk1−βkq2yk,Lyk≤q2yk,ω−4cMβk1−βkq2yk,Lyk.
Thus,(16)q2yk+1,ω≤1−αyq2Lyk,ω+αyq2Lwk,ω−4cMαk1−αkq2Lyk,Lwk≤1−αkq2yk,ω+αkq2wk,ω−4cMαk1−αkq2Lyk,Lwk≤1−αkq2yk,ω+αkq2wk,ω−4cMαk1−αkq2Lyk,Lwk≤1−αkq2yk,ω+αkq2wk,ω≤q2yk,ω−4cMαkβk1−βkq2yk,Lyk.
SincecM>0, it follows that
(17)∑k=1∞a21−bq2yk,Lyk≤∑k=1∞αkβk1−βkq2yk,Lyk<∞.
Thus,limk⟶∞q2yk,Lyk=0 and hence
(18)limk⟶∞qyk,Lyk=0.In some sense, the notion ofΔ-convergence in nonlinear domains provides the analog of the notion of weak convergence in linear setting. To obtain the Δ-convergence for operators having the condition E in a 2-UCHS by using the Agarwal iterative sequence (6), we shall propose the following techniques.Theorem 14.
Choose a 2-UCHSZ such that ∅≠S⊆Z is closed as well as convex. Furthermore, let L:S⟶S be endowed with condition E and FL≠∅. If yk is the Agarwal iterates (6) with restriction αk,βk∈a,b⊂0,1, then, ykΔ-converges to a fixed point of L.Proof.
It has been observed in Theorem13 that the iterative sequence yk is bounded having limk⟶∞qLyk,yk=0. We may consider the ωΔyk=⋃Auk and notice that the union has been imposed on all possible subsequences uk of the iterative sequence yk. We want to show that ωΔyk⊆FL. Suppose u∈ωΔyk. Then, one can find a subsequence uk of yk in such a way that Auk=u. By Lemmas 8 and 9, one can find a subsequence vk of uk in such a way that Δ−limk⟶∞vk=v∈S. Keeping limk⟶∞ρvk,Lvk=0 in mind and applying condition E of L, one has
(19)qvk,Lv≤μqvk,Lvk+qvk,v.
If we applylimsup on both of the sides of above, then, we have v∈FL. By Lemma 12, limk⟶∞qyk,v exists. Next, we show that u=v. Assume on contrary that u≠v. Then, using the property of uniqueness of asymptotic centers, we get
(20)limsupk⟶∞qvk,v=limsupk⟶∞qvk,u≤limsupk⟶∞quk,u<limsupk⟶∞quk,v=limsupk⟶∞qyk,v=limsupk⟶∞qvk,v.
The above strict inequality suggests a contradiction. So we must haveu=v∈FL and ωΔyk⊆FL.
Finally, we show thatyk is Δ-convergent in FL, that is, we want to show ωΔyk is singleton. Assume that uk is a subsequence of the sequence yk. In the view of Lemmas 8 and 9, one can find a subsequence vk of uk in such a way that Δ-limk⟶∞vk=v∈S. Suppose Auk=u and Ayk=x. We have already established that u=v and v∈FL. Now, we have to show that x=v. Suppose not, then since limk⟶qyk,v exists and also the asymptotic centers are consist of exactly one point, so
(21)limsupk⟶∞qvk,v=limsupk⟶∞qvk,x≤limsupk⟶∞qyk,x<limsupk⟶∞qyk,v=limsupk⟶∞qvk,v.which is obviously a contradiction and so x=v∈FL. Hence, ωΔyk=x.☐Normally, we are interested in a strong convergence. To obtain the strong convergence theorems, we essentially impose some other assumptions on the domain or the operator. We first take a compact domain and establish a strong convergence theorem as follows.Theorem 15.
Choose a 2-UCHSZ such that ∅≠S⊆Z is compact as well as convex. Furthermore, let L:S⟶S be endowed with condition E and FL≠∅. If yk is the Agarwal iterates (6) with restriction αk,βk∈a,b⊂0,1. Then, yk converges strongly to a fixed point of L.Proof.
SinceS is compact, we may select a subsequence yki of yk such that yki⟶g∈S. By using condition E of L, we have
(22)qyki,Lg≤μqyki,Lyki+qyki,g.
In the view of Theorem13, limk⟶∞qyki,Lyki=0. Hence, if we apply k⟶∞ on both of the sides of the above inequality, we get yki⟶Lg by uniqueness of limits in metric spaces, g=Lg. Hence, g is the fixed point of L. Moreover, by Lemma 12, limk⟶∞qyk,g exists. So yk is strongly convergent to g. This completes the proof.☐The compactness of the domain in the above theorem provided us a significant help. Now, one thinks about how to replace the compactness of the domain by any other assumption. We replace it by the following condition, which was essentially defined by Senter and Dotson [21].Definition 16.
A self-mapL on a subset S of a 2-UCHS Z is said to satisfy condition I, if there exists some nondecreasing function η:0,∞⟶0,∞ satisfying η0=0, ηa>0 for every a>0 and qs,Ls≥ηinfω∈FLqs,ω for each s∈S.Theorem 17.
Choose a 2-UCHSZ such that ∅≠S⊆Z is closed as well as convex. Furthermore, let L:S⟶S be endowed with conditions E and I, and FL≠∅. If yk is the Agarwal iterates (6) with restriction αk,βk∈a,b⊂0,1. Then, yk converges strongly to a fixed point of L.Proof.
Theorem13 gives limk⟶∞qyk,Lyk=0. By condition I, one has
(23)limk⟶∞qyk,FL=0.
Thus, one can choose a subsequenceykj of yk and ωj in FL such that qykj,ωj≤1/2j for every natural number j. Proof of Lemma 12 suggests that yk is nonicreasing, and so
(24)qykj+1,ωj≤qykj,ωj≤12j.
Now,(25)qωj+1,ωj≤qωj+1,ykj+1+ρykj+1,ωj≤12j+1+12j≤12j−1⟶0,asj⟶∞.
From the above, one can easily conclude that the sequenceωj form a Cauchy sequence in the closed subset FL of S. Hence, ωj⟶g∈FL and so g is a fixed point of L. But Lemma 12 suggests that limk⟶∞qyk,g exists. Thus, the element g is the strong limit point of the sequence yk.☐


## 3. Numerical Observations

This section is devoted to some numerical computations. We first suggest a nontrivial example ofL endowed with the condition E, but not with C. This example shows that the class of Garcia-Falset is properly more general than the class of Suzuki mappings.Example 1.
SupposeS=0,1 and L is a self-map defined as follows:
(26)Ls=0,ifs∈S1=0,15,5s6,ifs∈S2=15,1.We fix the value ofμ=6. We shall prove that qs,Ls′≤6qs,Ls+qs,s′ for each s,s′∈S.Case I.
Assume thats,s′∈S1. Then, Ls=Ls′=0. Thus,
(27)qs,Ls′=s−Ls′=s≤6s=6s−Ls≤6s−Ls+s−s′=6qs,Ls+qs,s′.Case II.
Here, we chooses,s′∈S2. Then Ls=5s/6 and Ls′=5s′/6. Thus,
(28)qs,Ls′≤qs,Ls+qLs,Ls′=s−Ls+Ls−Ls′=s−Ls+5s6−5s′6=s−Ls+56s−s′≤s−Ls+s−s′≤6s−Ls+s−s′=6qs,Ls+qs,s′.Case III.
Ifs∈S1 and s′∈S2. Then Ls=0 and Ls′=5s′/6. So,
(29)qs,Ls′=s−Ls′=s−5s′6=6s−5s′6=s+5s−5s′6≤s6+5s−5s′6≤s+s−s′=s−Ls+s−s′≤6s−Ls+s−s′=6qs,Ls+qs,s′.Case IV.
Ifs∈S2 and s′∈S1. Then Ls=5s/6 and Ls′=0 and so
(30)qs,Ls′=s−Ls′=s=6s6=6s−Ls≤6s−Ls+s−s′=6qs,Ls+qs,s′.Next, we provide the values for whichL does not satisfy condition C. Choose s=1/8 and s′=1/5. It is easy to show 1/2ds,Ls<qs,s′, but qLs,Ls′>qs,s′. For any choice of k∈ℕ, let αk=0.70 and βk=0.65, then, the observations are given in Table 1.Table 1
Strong convergence of Agarwal, Ishikawa, and Mann iterative schemes.
Agarwal (6)Ishikawa (5)Mann (4)y10.90.90.9y20.6931250000000000.738125000000000.79499999999999y30.5338025173611110.605365017361110.70224999999999y40.4111020776065770.496483392710740.62032083333333y50.3166056972678430.407185338049570.54795006944444y60.2438303599097480.333948530719820.48402000000000y70.1877832424582710.273884176930630.42755326251928y800.224623064552130.37767204855870y900.184222110580600.33361030956019y1000.055266633174180.29468910677816Now, we study the behavior of Agarwal, Ishikawa, and Mann iterations under different cases. Notice thatqyk,ω<10−10 is the stopping criteria, y1=0.9, and we stop the iterates for k=50. In this case, Figures 1, 2, and 3 show that the Agarwal iteration process converges faster to the fixed point ω=0 as compared to the other iterations.Figure 1
αk=1/k+1, βk=k/k+1.Figure 2
αk=1−1/6k+1, βk=1/k3.Figure 3
αk=1/2, βk=1/k+1.


## 4. Conclusions

In this paper, we have initiated the finding of fixed points for the generalized class of mappings due to Garcia-Falset et al. [10] in the general setting of 2-UCHS. We established several strong and Δ-convergence theorems under the Agarwal iterative process [8]. We have presented a new example of mapping having condition E but not C and proved that its Agarwal iterative process [8] is more effective than the corresponding Mann [13] and Ishikawa [12] iterative processes. In view of the above discussion, our results simultaneously hold in UCBS, CAT(0), and some CAT(κ) spaces. The present results can be extended and applied to many fields of applied sciences and differential equations. Thus, our results extend the corresponding results of Bagherboum [11], Phuengrattana [7], Basarir and Sahin [9], and Nanjaras et al. [6] from the general setting of domains and faster iteration process.



---

*Source: 7930128-2021-07-09.xml*