Iterative Construction of Fixed Points for Operators Endowed with Condition ðEÞ in Metric Spaces

Department of Mathematics and Statistics, International Islamic University, H-10 Islamabad-44000, Pakistan Department of Mathematics, University of Lakki Marwat, Lakki Marwat, 28420 Khyber Pakhtunkhwa, Pakistan Department of Engineering Science, Bandırma Onyedi Eylül University, 10200 Bandırma, Balıkesir, Turkey Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644Bilbao, Barrio Sarriena, 48940 Leioa, Spain


Introduction
Choose a metric space Z and ∅≠ S ⊆ Z. Then, a self-map L : S ⟶ S is called nonexpansive if q Ls, Ls′ ≤ q s, s′ , for all s, 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 ðCÞ if any given s, s′ ∈ S, Obviously, a nonexpansive map L 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 selfmap L of S is called Garcia-Falset nonexpansive or said to be endowed with the condition ðEÞ if any given s, s ′ ∈ S, there is a μ ≥ 1 such that Remark 1. If L is nonexpansive, then it is easy to show qðs, Ls ′ Þ ≤ qðs, LsÞ + qðs, s ′ Þ for each s, s ′ ∈ S. Hence, every nonexpansive map L has condition ðEÞ with μ = 1. Similarly, if L is Suzuki map, then, it satisfies qðs, Ls′Þ ≤ 3qðs, LsÞ + qð s, s′Þ for every s, s′ ∈ S. Hence, every Suzuki map L has condition ðEÞ with μ = 3. Interestingly, the example constructed in this paper shows that there exists maps satisfying condition ðEÞ 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 iteration y k+1 = Ly k 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: where α k ∈ ½0, 1. The Mann iteration (4) was extended to the setting of two steps by Ishikawa [12] as follows: where α k , β k ∈ ½0, 1.
In 2007, Agarwal et al. [8] studied the following iteration for contractions and observed that it is better than the above iterative processes: where α k , β k ∈ ½0, 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 2uniformly convex hyperbolic metric space for establishing the main outcome. To support our results, we provide an example of maps with condition ðEÞ 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 4. Suppose a pair ðZ, qÞ is a given hyperbolic space. For any r ∈ ð0,∞Þ and ε ∈ ð0, 2, define: and keeping in mind that infimum is taken over every s, s ′ , p ∈ Z satisfying qðs, pÞ ≤ r, qðs ′ , pÞ ≤ r, and qðs, s ′ Þ ≥ rε. Then, ðZ, qÞ is called 2-uniformly convex hyperbolic space 2 Advances in Mathematical Physics The following remark suggests that 2-UCHS is a more general domain than many other linear and nonlinear spaces.
Definition 7. Choose a complete 2-UCHS Z such that ∅ ≠ S ⊆ Z is closed as well as convex. Fix an element w ∈ S . Assume that fy k g is a bounded sequence in Z. Then, fy k g is called Δ-convergent to w iff the asymptotic center AðS, fz k gÞ = fwg for all subsequence fz k g of fy k g. When w is Δ-limit of fy k g, then we shall write Δ − lim k⟶∞ y k = w.
Lemma 9 (see [20]). Choose a complete 2-UCHS Z endowed with MMUC such that ∅≠ S ⊆ Z and fy k g ⊆ S is bounded. Then, the asymptotic center of fy k g is always in S.
Mappings endowed with condition ðEÞ enjoy the following useful properties which we have combined in a proposition form. For details, we refer the reader to [10].
(i) When the map L has condition ðEÞ with nonempty fixed point set, then for each s ∈ S and ω ∈ F L , we have qðLs, LωÞ ≤ qðs, ωÞ (ii) When the map L has condition ðEÞ then fixed point set of S is always closed in S We now state an important characterization of a 2-UCHS, which was proved in [15]. Lemma 11. Choose a 2-UCHS ðZ, qÞ. Then, the following inequality holds for each ξ ∈ ½0, 1 and s, s ′ , g ∈ Z.

Approximation Results
We now want to show our desirable strong and Δ-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 F L 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-UCHS Z such that ∅≠ S ⊆ Z is closed as well as convex. Furthermore, let L : S ⟶ S be endowed with condition ðEÞ having F L ≠ ∅. If fy k g is the Agarwal iterates (6), then, lim k⟶∞ qðy k , ωÞ exists for all ω ∈ F L .
Proof. Assume that ω ∈ F L . By using Proposition 10 (i), one has Hence, From the above observations, we have qðy k+1 , ωÞ ≤ qðy k , ωÞ. Hence, the real sequence fqðy k , ωÞg is bounded below by 0 and nonincreasing as well, so lim k⟶∞ qðy k , ωÞ exists for every ω ∈ F L .
The existence of a fixed point for a self-map L : S ⟶ S having condition ðEÞ 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-UCHS Z such that ∅≠ S ⊆ Z is closed as well as convex. Furthermore, let L : S ⟶ S be endowed with condition ðEÞ. If fy k g is the Agarwal iterates (6) with restriction α k , β k ∈ ½a, b ⊂ ð0, 1Þ. Then, F L ≠ ∅ iff f y k g is bounded and lim k⟶∞ qðy k , Ly k Þ = 0.

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Proof. Suppose that the iterative sequence fy k g is bounded having lim k⟶∞ qðLy k , y k Þ = 0. We may fix an element, namely, ω ∈ AðS, fy k gÞ. By using condition ðEÞ of L, we have It follows that Lω ∈ AðS, fy k gÞ. Since AðS, fy k gÞ is singleton set, one has Lω = ω. Thus, F L ≠ ∅. Conversely, we suppose that F L ≠ ∅ and ω ∈ F L . Then, by the proof of Lemma 12, fy k g is bounded. By Lemma 11, we have Thus, Since c M > 0, it follows that Thus, lim k⟶∞ q 2 ðy k , Ly k Þ = 0 and hence lim k⟶∞ q y k , Ly k ð Þ= 0: In some sense, the notion of Δ-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 ðEÞ in a 2-UCHS by using the Agarwal iterative sequence (6), we shall propose the following techniques.

Theorem 14.
Choose a 2-UCHS Z such that ∅≠ S ⊆ Z is closed as well as convex. Furthermore, let L : S ⟶ S be endowed with condition ðEÞ and F L ≠ ∅. If fy k g is the Agarwal iterates (6) with restriction α k , β k ∈ ½a, b ⊂ ð0, 1Þ, then, f y k gΔ-converges to a fixed point of L.
Proof. It has been observed in Theorem 13 that the iterative sequence fy k g is bounded having lim k⟶∞ qðLy k , y k Þ = 0. We may consider the ω Δ ðy k Þ = S Aðfu k gÞ and notice that the union has been imposed on all possible subsequences f u k g of the iterative sequence fy k g. We want to show that ω Δ ðy k Þ ⊆ F L . Suppose u ∈ ω Δ ðy k Þ. Then, one can find a subsequence fu k g of fy k g in such a way that Aðfu k gÞ = fug. By Lemmas 8 and 9, one can find a subsequence fv k g of fu k g in such a way that Δ − lim k⟶∞ v k = v ∈ S. Keeping lim k⟶∞ ρð v k , Lv k Þ = 0 in mind and applying condition ðEÞ of L, one has If we apply limsup on both of the sides of above, then, we have v ∈ F L . By Lemma 12, lim k⟶∞ qðy k , vÞ exists. Next, we show that u = v. Assume on contrary that u ≠ v. Then, using the property of uniqueness of asymptotic centers, we get The above strict inequality suggests a contradiction. So we must have u = v ∈ F L and ω Δ ðy k Þ ⊆ F L .
Finally, we show that fy k g is Δ-convergent in F L , that is, we want to show ω Δ ðy k Þ is singleton. Assume that fu k g is a subsequence of the sequence fy k g. In the view of Lemmas 8 and 9, one can find a subsequence fv k g of fu k g in such a way that Δ-lim k⟶∞ v k = v ∈ S. Suppose Aðfu k gÞ = fug and Aðfy k gÞ = fxg. We have already established that u = v and v ∈ F L . Now, we have to show that x = v. Suppose not, then since lim k⟶ qðy k , vÞ exists and also the asymptotic centers are consist of exactly one point, so which is obviously a contradiction and so x = v ∈ F L . Hence, ω Δ ðy k Þ = fxg. ☐ 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-UCHS Z such that ∅≠ S ⊆ Z is compact as well as convex. Furthermore, let L : S ⟶ S be endowed with condition ðEÞ and F L ≠ ∅. If fy k g is the

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Agarwal iterates (6) with restriction α k , β k ∈ ½a, b ⊂ ð0, 1Þ. Then, fy k g converges strongly to a fixed point of L.
Proof. Since S is compact, we may select a subsequence fy k i g of fy k g such that y k i ⟶ g ∈ S. By using condition ðEÞ of L, we have In the view of Theorem 13, lim k⟶∞ qðy k i , Ly k i Þ = 0. Hence, if we apply k ⟶ ∞ on both of the sides of the above inequality, we get y k i ⟶ Lg by uniqueness of limits in metric spaces, g = Lg. Hence, g is the fixed point of L. Moreover, by Lemma 12, lim k⟶∞ qðy k , gÞ exists. So fy k g 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-map L on a subset S of a 2-UCHS Z is said to satisfy condition ðIÞ, if there exists some nondecreasing function η : ½0,∞Þ ⟶ ½0,∞Þ satisfying ηð0Þ = 0, ηðaÞ > 0 for every a > 0 and qðs, LsÞ ≥ ηðinf ω∈F L qðs, ωÞÞ for each s ∈ S . Theorem 17. Choose a 2-UCHS Z such that ∅≠ S ⊆ Z is closed as well as convex. Furthermore, let L : S ⟶ S be endowed with conditions ðEÞ and ðIÞ, and F L ≠ ∅. If fy k g is the Agarwal iterates (6) with restriction α k , β k ∈ ½a, b ⊂ ð0 , 1Þ. Then, fy k g converges strongly to a fixed point of L.
Proof. Theorem 13 gives lim k⟶∞ qðy k , Ly k Þ = 0. By condition ðIÞ, one has Thus, one can choose a subsequence fy k j g of fy k g and fω j g in F L such that qðy k j , ω j Þ ≤ 1/2 j for every natural number j. Proof of Lemma 12 suggests that fy k g is nonicreasing, and so Now, From the above, one can easily conclude that the sequence fω j g form a Cauchy sequence in the closed subset F L of S. Hence, ω j ⟶ g ∈ F L and so g is a fixed point of L. But Lemma 12 suggests that lim k⟶∞ qðy k , gÞ exists. Thus, the element g is the strong limit point of the sequence fy k g. ☐

Numerical Observations
This section is devoted to some numerical computations. We first suggest a nontrivial example of L endowed with the condition ðEÞ, but not with ðCÞ. This example shows that the class of Garcia-Falset is properly more general than the class of Suzuki mappings.

So,
Case IV. If s ∈ S 2 and s ′ ∈ S 1 . Then Ls = 5s/6 and Ls ′ = 0 and Next, we provide the values for which L does not satisfy condition ðCÞ. Choose s = 1/8 and s′ = 1/5. It is easy to show 1/2dðs, LsÞ < qðs, s′Þ, but qðLs, Ls′Þ > qðs, s′Þ. For any choice of k ∈ ℕ, let α k = 0:70 and β k = 0:65, then, the observations are given in Table 1. Now, we study the behavior of Agarwal, Ishikawa, and Mann iterations under different cases. Notice that qðy k , ωÞ < 10 −10 is the stopping criteria, y 1 = 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.

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 ðEÞ but not ðCÞ 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], Phuengrat-tana [7], Basarir and Sahin [9], and Nanjaras et al. [6] from the general setting of domains and faster iteration process.

Data Availability
The data used to support the findings of this study are included in the references within the article.

Conflicts of Interest
The authors declare no conflict of interest.