Int. J. Anal. Appl. (2023), 21:62 Strong and ∆-Convergence of a New Iteration for Common Fixed Points of Two Asymptotically Nonexpansive Mappings J. Robert Dhiliban1,∗, A. Anthony Eldred2 1Department of Mathematics, Arul Anandar College (Autonomous), Madurai-625514, Tamilnadu, India 2Department of Mathematics, St. Joseph’s College (Autonomous), Affiliated to Bharathidasan University, Tiruchirappalli-620002, Tamilnadu, India ∗Corresponding author: jrdhiliban@gmail.com Abstract. The purpose of this paper is to study strong and ∆ - convergence of a newly defined iteration to a common fixed point of two asymptotically nonexpansive self mappings in a hyperbolic space framework. We provide an example and a comparison table to support our assertions. 1. Introduction Globel and Kirk [1] introduced the concept of asymptotically nonexpansive mappings and proved that every asymptotically nonexpansive self mapping on a non empty closed subset K of a uniformly convex Banach space X posseses a fixed point. Ever since, many authors (see, [2], [3], [4] and [5]) have established strong and weak convergence theorems for asymptotically nonexpansive mappings based on the modified Mann [6] and Ishikawa [7] iterations. Tan and Xu [8] studied the modified Ishikawa iteration scheme:   x1 ∈ K xn+1 = (1 −αn)xn + αnTnyn yn = (1 −βn)xn + βnTnxn, n ≥ 1 (1.1) where {αn} and {βn} are sequences in (0, 1) bounded away from 0 and 1. Received: Apr. 28, 2023. 2020 Mathematics Subject Classification. 47H10, 47J25. Key words and phrases. asymptotically nonexpansive mapping; common fixed points; ∆-convergence; hyperbolic space. https://doi.org/10.28924/2291-8639-21-2023-62 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-62 2 Int. J. Anal. Appl. (2023), 21:62 Aggarwal et al [9] in an attempt to obtain a faster rate of convergence, modified the above iteration process (1.1) as following:   x1 ∈ K xn+1 = (1 −αn)Tnxn + αnTnyn yn = (1 −βn)xn + βnTnxn, n ≥ 1 (1.2) This iteration is called the modified S-iteration process. For further results on Ishikawa iteration process, (refer, [10], [11], [12] and [13]). Recently, iterative approximations are defined and inves- tigated in the framework of hyperbolic spaces. Several authors (refer, [14], [15] and [16]) have put forward different notions of hyperbolic spaces in order to blend convexity and metric structure. The following definition given by Kohlenbach [17] is widely used. Definition 1.1. [17] A hyperbolic space is a triplet (X,d,W ), where (X,d) is a metric space and W : X2 → [0, 1] is a mapping that satisfies the following conditions: (1) d(u,W (x,y,α)) ≤ (1 −α)d(u,x) + αd(u,y) (2) d(W (x,y,α),W (x,y,β)) = |α−β|d(x,y) (3) W (x,y,α) = W (y,x, (1 −α)) (4) d(W (x,z,α),W (y,v,α)) ≤ (1 −α)d(x,y) + αd(z,v) for all x,y,z,u,v ∈ X and α,β ∈ [0, 1]. 2. Preliminaries We recall some definitions and basic concepts which will be useful for our work. Definition 2.1. [1] Let (X,d) be a metric space and let K be a closed convex subset of X. A mapping T : K → K is said to be asymptotically nonexpansive, if there is a sequence of real numbers {kn}∈ [1,∞) such that lim n→∞ kn = 1 and d(Tnx,Tny) ≤ knd(x,y) for all x,y ∈ X and ∀ n ∈N. The concept of an asymptotically nonexpansive mapping is a natural generalization of a nonexpan- sive mapping (d(Tx,Ty) ≤ d(x,y)). The set F (T ) = {Tx = x : x ∈ K} shall denote the set of all fixed points of any mapping T. Definition 2.2. [23] A subset K of a hyperbolic space (X,d,W ) is convex if W (x,y,α) ∈ K for all x,y ∈ K and ∀ α ∈ [0, 1]. Definition 2.3. [24] A hyperbolic space (X,d,W ) is said to be uniformly convex if for any x,y,z ∈ X, r > 0 and � ∈ (0, 2], there is a δ ∈ (0, 1] so that d(W (x,y, 1 2 ),z) ≤ (1 − δ)r whenever d(x,z) ≤ r, d(y,z) ≤ r and d(x,y) ≥ �r. Definition 2.4. [25], [26] Consider a bounded sequence {xn} in a hyperbolic space (X,d,W ). For any x ∈ X, define, r(x,{xn}) = lim n→∞ sup d(xn,x) and r({xn}) = inf{r(x,{xn})/x ∈ X}. The Int. J. Anal. Appl. (2023), 21:62 3 asymptotic center A({xn}) of a bounded sequence {xn} is defined as A({xn}) = {x ∈ X/r(x,{xn}) ≤ r(y,{xn}),∀ y ∈ X}. It is well known that in uniformly convex Banach spaces, bounded sequences have unique asymptotic centers. The following Lemma proved by Leustean [27] guarantees that complete uniformly convex hyperbolic spaces also enjoy this property. Lemma 2.1. [27] Let (X,d,W ) be a complete uniformly convex hyperbolic space. Then every bounded sequence {xn} in X has a unique asymptotic center. Definition 2.5. [28] A sequence {xn} in a hyperbolic space (X,d,W ) is said to ∆-converge to a point x ∈ X, if every subsequence {zn} of {xn} has x as its unique asymptotic center. Lemma 2.2. [29] Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space (X,d,W ) and let {xn} be a bounded sequence in K such that A({xn}) = {z} and r({xn}) = ω. If {zm} is a sequence in K such that lim m→∞ r(zm,{xn}) = ω, then lim m→∞ zm = z. Lemma 2.3. [29] Let (X,d,W ) be a uniformly convex hyperbolic space. Let x ∈ X and let {tn} be a sequence in (0, 1) such that δ ≤ tn ≤ 1 −δ for all n ∈N and for some δ > 0. If {xn} and {yn} are sequences in X such that lim n→∞ sup d(xn,x) ≤ c, lim n→∞ sup d(yn,x) ≤ c and lim n→∞ d(W (xn,yn,tn),x) = c for some c ≥ 0, then lim n→∞ d(xn,yn) = 0. Lemma 2.4. [3] Let {αn}, {βn} and {δn} be sequences of nonnegative numbers such that δn+1 ≤ αnδn + βn ∀ n ∈N. If αn ≥ 1 ∀ n ∈N and ∑∞ n=1(αn − 1) < ∞ and βn < ∞, then limn→∞δn exists. Uniformly convex Banach spaces and CAT(0) spaces are some of the known examples of hyperbolic spaces. Sahin and Basarir [18] studied the following iterative process in a hyperbolic space setting and established some convergence results under suitable conditions:  x1 ∈ K xn+1 = W (T nxn,T nyn,αn) yn = W (xn,T nxn,βn), n ≥ 1 (2.1) Ishikawa type iteration is also employed to study the convergence of common fixed points of two asymptotically nonexpansive mappings. In a Banach space framework, Das and Debata [19] initiated the study of two mapping iterative procedure. The authors in [20] and [21] have studied the following iteration for the convergence of common fixed points:  x1 ∈ K xn+1 = W (xn,S nyn,αn) yn = W (xn,T nxn,βn), n ≥ 1 (2.2) 4 Int. J. Anal. Appl. (2023), 21:62 where S and T are asymptotically nonexpansive mappings with atleast one common fixed point and {αn} and {βn} are sequences in (0, 1). Recently, Saluja [22] modified the iterative procedure introduced by Khan et al [13] in hyperbolic spaces to obtain a faster iterative procedure:  x1 ∈ K xn+1 = W (T nxn,S nyn,αn) yn = W (xn,T nxn,βn), n ≥ 1 (2.3) where S and T are asymptotically nonexpansive mappings with atleast one common fixed point and {αn} and {βn} are sequences in (0, 1). The purpose of this paper is to introduce and study a new iterative procedure (3.1) even in Banach spaces to approximate the common fixed points of two asymptotically nonexpansive mappings. We prove strong and ∆ - convergence of such an iteration in the general nonlinear framework of hyperbolic spaces. 3. Main Results In this section, we introduce and study a new iterative scheme to approximate common fixed points of two asymptotically nonexpansive mappings in a hyperbolic space. Let (X,d,W ) be a uniformly convex hyperbolic space. Let K be a non-empty subset of X. Let S and T be two asymptotically nonexpansive self mappings on K. Let {αn} and {βn} be sequences in (0, 1) such that, δ ≤ αn, βn ≤ 1 −δ, for all n ∈N and for some δ > 0. We define the following iteration:  x1 = x ∈ K xn+1 = W (S nxn,T nyn,αn) yn = W (xn,S n(Tnxn),βn), n ≥ 1 (3.1) Lemma 3.1. Let K be a non-empty subset of a uniformly convex hyperbolic space X. Let S and T be asymptotically nonexpansive self mappings on K with a common sequence of real numbers kn ≥ 1 satisfying ∑ (k2n − 1) < ∞. Let F denote the set of all common fixed points of S and T. i.e., F = F (S) ∩F (T ). Let p ∈ F. If {xn} and {yn} are sequences as defined in (3.1), then lim n→∞ d(xn,p) and lim n→∞ d(yn,p) exist and lim n→∞ d(xn,p) = lim n→∞ d(yn,p). Proof. Since p ∈ F (S) ∩F (T ), d(xn+1,p) = d(W (S nxn,T nyn,αn),p) ≤ (1 −αn)d(Snxn,p) + αnd(Tnyn,p) Int. J. Anal. Appl. (2023), 21:62 5 ≤ (1 −αn)knd(xn,p) + αnknd(yn,p) (3.2) d(yn,p) = d(W (xn,S n(Tnxn),βn),p) ≤ (1 −βn)d(xn,p) + βnd(Sn(Tnxn),p) = (1 −βn)d(xn,p) + βnd(Sn(Tnxn),Sn(Tnp)) ≤ (1 −βn)d(xn,p) + βnknd(Tnxn,Tnp) ≤ (1 −βn)d(xn,p) + βnk2nd(xn,p) = d(xn,p) [ (1 −βn) + βnk2n ] (3.3) By substituting (3.3) in (3.2), we get, d(xn+1,p) ≤ (1 −αn)knd(xn,p) + αnkn [ (1 −βn) + βnk2n ] d(xn,p) = [ (1 −αn)kn + αnkn((1 −βn) + βnk2n) ] d(xn,p) = [ kn −αnknβn + αnβnk3n ] d(xn,p) = [ 1 + (kn − 1) −αnβnkn + αnβnk3n ] d(xn,p) = [ 1 + (kn − 1) + (k2n − 1)αnβnkn ] d(xn,p) (3.4) Hence, d(xn+1,p) ≤ [ 1 + (kn − 1) + (k2n − 1)αnβnkn ] d(xn,p) By Lemma 2.4, lim n→∞ d(xn,p) exists. Let lim n→∞ d(xn,p) = c. (3.5) From (3.2), we have, d(yn,p) ≤ [ (1 −βn) + βnk2n ] d(xn,p) Hence, lim n→∞ sup d(yn,p) ≤ lim n→∞ sup d(xn,p) i.e., lim n→∞ sup d(yn,p) ≤ c. (3.6) Now consider, d(xn+1,p) = d(W (S nxn,T nyn,αn),p) ≤ (1 −αn)knd(xn,p) + αnknd(yn,p) = [ 1 + (kn − 1) + (k2n − 1)αnβnkn ] d(xn,p) By (3.5), we have, lim n→∞ sup d(xn+1,p) = c and lim n→∞ sup d(xn,p) = c. Hence, from (3.2) and (3.4), lim n→∞ sup [ (1 −αn)knd(xn,p) + αnknd(yn,p) ] = c. 6 Int. J. Anal. Appl. (2023), 21:62 i.e, lim n→∞ sup [ knd(xn,p) −knαnd(xn,p) + αnknd(yn,p) ] = c. Since, lim n→∞ sup kn = 1, we have, c + lim n→∞ sup αnkn [ d(yn,p) −d(xn,p) ] = c =⇒ lim n→∞ sup αnkn [ d(yn,p) −d(xn,p) ] = 0. Since, lim n→∞ sup αnkn > 0, this will imply that, lim n→∞ sup [ d(yn,p) −d(xn,p) ] = 0. Therefore, lim n→∞ sup d(yn,p) = c. Similarly, we can show that, lim n→∞ inf d(yn,p) = c. Hence, lim n→∞ d(yn,p) = c (3.7) � Lemma 3.2. Let K be a non-empty subset of a uniformly convex hyperbolic space X. Let S and T be asymptotically nonexpansive self mappings on K with a common sequence of real numbers kn ≥ 1 satisfying ∑ (k2n − 1) < ∞. If {xn} is a sequence as defined in (3.1) and d(xn,xn+1) → 0 as n →∞, then lim n→∞ d(xn,Sxn) = 0 and lim n→∞ d(xn,Txn) = 0. Proof. Let F denote the set of all common fixed points of S and T . i.e., F = F (S) ∩F (T ). Let p ∈ F. Now since, lim n→∞ kn = 1, from (3.5), we have, lim n→∞ sup d(Tnyn,p) ≤ lim n→∞ sup d(xn,p) = c. Similarly, lim n→∞ sup d(Snxn,p) ≤ c. (3.8) Now, d(xn+1,p) = d(W (S nxn,T nyn,αn),p) ≤ [1 + (kn − 1) + (k2n − 1)αnβnkn]d(xn,p). From (3.5), we have, d(W (Snxn,Tnyn,αn),p) = c. By Lemma 2.3, we have, lim n→∞ d(Snxn,T nyn) = 0. (3.9) Int. J. Anal. Appl. (2023), 21:62 7 Now consider, d(yn,p) = d(W (xn,S n(Tnxn),βn),p) ≤ (1 −βn)d(xn,p) + βnd(Sn(Tnxn),p) = d(xn,p) [ (1 −βn) + βnk2n ] . Since, lim n→∞ sup d(yn,p) = c and lim n→∞ sup d(xn,p) = c, we have, d(W (xn,S n(Tnxn),βn),p) → c Further, lim n→∞ sup d(Sn(Tnxn),p) ≤ c. (3.10) So, using Lemma 2.3, we conclude that, lim n→∞ d(xn,S n(Tnxn)) = 0. (3.11) Now, d(yn,xn) = d(W (xn,S n(Tnxn),βn),xn) ≤ (1 −βn)d(xn,xn) + βnd(Sn(Tnxn),xn). Using (3.11), lim n→∞ d(xn,yn) = 0. (3.12) From d(yn,S n(Tnxn)) ≤ d(yn,xn) + d(xn,Sn(Tnxn)), we have, lim n→∞ d(yn,S n(Tnxn)) = 0. (3.13) Now, d(xn+1,S nxn) = d(W (S nxn,T nyn,αn),S nxn) ≤ (1 −αn)d(Snxn,Snxn) + αnd(Tnyn,Snxn) ≤ (1 −αn)knd(xn,xn) + αnd(Tnyn,Snxn). So, lim n→∞ d(xn+1,S nxn) = 0. (3.14) Further, d(xn+1,T nyn) = d(W (S nxn,T nyn,αn),T nyn) ≤ (1 −αn)d(Snxn,Tnyn) + αnd(Tnyn,Tnyn) ≤ (1 −αn)d(Snxn,Tnyn) + αnknd(yn,yn) yields, lim n→∞ d(xn+1,T nyn) = 0. (3.15) Now, d(xn,S nxn) ≤ d(xn,xn+1) + d(xn+1,Snxn) So, lim n→∞ d(xn,S nxn) = 0. (3.16) 8 Int. J. Anal. Appl. (2023), 21:62 Now consider, d(xn,Sxn) ≤ d(xn,xn+1) + d(xn+1,Sn+1xn+1) + d(Sn+1xn+1,Sn+1xn) + d(Sn+1xn,Sxn) ≤ d(xn,xn+1) + d(xn+1,Sn+1xn+1) + kn+1d(xn+1,xn) + k1d(Snxn,xn). Thus, we conclude that, lim n→∞ d(xn,Sxn) = 0. (3.17) And from, d(xn,T nyn) ≤ d(xn,xn+1) + d(xn+1,Tnyn) we obtain, lim n→∞ d(xn,T nyn) = 0 (3.18) and therefore, lim n→∞ d(yn,T nyn) = 0. (3.19) Also, d(yn,yn+1) ≤ d(yn,xn) + d(xn,xn+1) + d(xn+1,yn+1). Thus, lim n→∞ d(yn,yn+1) = 0. (3.20) Now consider, d(yn,Tyn) ≤ d(yn,yn+1) + d(yn+1,Tn+1yn+1) + d(Tn+1yn+1,T n+1yn) + d(T n+1yn,Tyn) ≤ d(yn,yn+1) + d(yn+1,Tn+1yn+1) + kn+1d(yn+1,yn) + k1d(Tnyn,yn). Therefore, lim n→∞ d(yn,Tyn) = 0. (3.21) By the asymptotic nonexpansive property of T, d(Txn,Tyn) ≤ k1d(xn,yn). Hence, lim n→∞ d(Txn,Tyn) = 0. (3.22) From, d(xn,Txn) ≤ d(xn,yn) + d(yn,Tyn) + d(Tyn,Txn), we conclude that, lim n→∞) d(xn,Txn) = 0. This completes the proof. (3.23) � Theorem 3.1. Let K be a non-empty closed convex subset of a uniformly convex hyperbolic space (X,d,W ). Let T : K → K and S : K → K be asymptotically nonexpansive mappings with F (T ) 6= φ and F (S) 6= φ and kn ≥ 1 satisfying ∑∞ n=1(k 2 n − 1) < ∞. For any initial point x1 ∈ K, define the sequence {xn} iteratively by (3.1). Suppose d(xn,xn+1) → 0 as n → ∞, then, {xn} ∆-converges to an element of F (T ) ∩F (S). Int. J. Anal. Appl. (2023), 21:62 9 Proof. From Lemma 3.2, d(xn,Txn) → 0 and d(xn,Sxn) → 0 as n →∞. Lemma 2.1 ensures that any bounded sequence has a unique asymptotic center. Let {zn} be a subsequence of {xn}. Since {xn} is bounded, {zn} is also bounded and suppose that A({xn}) = x and A({zn}) = z. Using the asymptotic nonexpansive property of T, we have, lim n→∞ d(Tkzn,T k+1zn) = 0, where k = 1, 2, 3, ... Our purpose is to show that, z = x and z ∈ F (T ) ∩F (S). Let m and n be positive integers. Now, d(Tmz,zn) ≤ d(Tmz,Tmzn) + d(Tmzn,Tm−1zn) + ... + d(Tzn,zn) ≤ kmd(z,zn) + m−1∑ k=0 d(Tkzn,T k+1zn). Taking lim sup as n →∞, for any fixed m, we have, r(Tmz,{zn}) = lim n→∞ sup d(Tmz,{zn}) ≤ km lim n→∞ sup d(z,{zn}) = kmr(z,{zn}). Now, taking lim sup as m →∞, we obtain, lim m→∞ sup r(Tmz,{zn}) ≤ r(z,{zn}). Since A({zn}) = z, we have, r(z,{zn}) ≤ r(Tmz,{zn}), for any fixed m ∈ N, which implies that, lim m→∞ r(Tmz,{zn}) = r(z,{zn}). Using Lemma 2.2, we conclude that, Tmz → z and z ∈ F (T ). By a similar argument, we can show that z ∈ F (S). We now claim that, z is the unique asymptotic center for each subsequence {zn} of {xn}. Suppose x 6= z. Since z ∈ F (T ) ∩F (S), by Lemma 3.1, lim n→∞ d(xn,z) exists and therefore by the uniqueness of asymptotic centers, we have, lim n→∞ sup d(zn,z) < lim n→∞ sup d(zn,x) ≤ lim n→∞ sup d(xn,x) < lim n→∞ sup d(xn,z) = lim n→∞ sup d(zn,z). This contradiction proves that z must be equal to x. Since the choice of the subsequence {zn} is arbitrary, we have, A({zn}) = {x}, for all subsequences {zn} of {xn}. Thus, we conclude that, {xn} ∆-converges to a common fixed point of T and S. � Theorem 3.2. Let K be a non-empty subset of a uniformly convex hyperbolic space X. Let S and T be asymptotically nonexpansive self mappings on K. Let {xn} and {yn} be sequences as defined in 10 Int. J. Anal. Appl. (2023), 21:62 (3.1) and d(xn,xn+1) → 0 as n → ∞. If either of the mappings T or S is demi-compact, then {xn} and {yn} converge strongly to an element of F (T ) ∩F (S). Proof. Assume T is demi-compact. By Theorem 3.1, we have, d(xn,Txn) → 0 as n → ∞. Then, there exists a subsequence {xnp} of {xn} such that Txnp → z∗. Now, d(xnp,z ∗) ≤ d(xnp,Txnp) + d(Txnp,z∗) → 0 as p →∞. Since, lim n→∞ d(xn,Txn) → 0, we have z∗ ∈ F (T ). Also, lim n→∞ d(xn,z ∗) exists. Hence, xn → z∗ and d(xn,yn) → 0 implies that lim n→∞ d(yn,z ∗) exists. Further, d(xn,Sxn) → 0 implies that z∗ ∈ F (S). Hence, {xn} and {yn} converges strongly to z∗ ∈ F (T ) ∩F (S). � As an illusration, we consider the following example in a Banach space setting. Example 3.1. Consider K = B(0; 0.9) , the ball centred at 0 and radius 0.9 in R2. Let S and T be self mappings on K defined by S(x1,x2) = (x21,x 2 2) and T (x1,x2) = (sin x1, sin x2). Let x,y ∈ K, so that x = (x1,x2) and y = (y1,y2). Assume that y1 < x1 and y2 < x2. Now, d(Snx,Sny) = ‖Snx −Sny‖ = ∥∥(x2n1 ,x2n2 )−(y2n1 ,y2n2 )∥∥ = [( x2n1 −y 2n 1 )2 + ( x2n2 −y 2n 2 )2]12 = [ |x1 −y1|2 { x2n−11 + y1x 2n−2 1 + ... + y 2n−1 1 }2 + |x2 −y2|2 { x2n−12 + y2x 2n−2 2 + ... + y 2n−1 2 }2]12 ≤ [ |x1 −y1|2 { 2nx2n−11 }2 + |x2 −y2|2 { 2nx2n−12 }2]12 Take ln = max { 1, 2nx2n−11 } and mn = max { 1, 2nx2n−12 } . Let kn = max{ln,mn}. Then clearly kn → 1 as n →∞. So, d(Snx,Sny) ≤ kn [ |x1 −y1|2 + |x2 −y2|2 ]1 2 = kn‖x −y‖. Hence S is an asymptotically nonexpansive mapping on K. Also T is a nonexpansive mapping on K and (0, 0) is a common fixed point of T and S. The following table shows that our new iterative scheme has a comparitively better rate of convergence than some of the existing iterative schemes. Here, we take x1 = ( 3 4 , 3 4 ) and αn = βn = 1 2 ,∀n ∈N. Int. J. Anal. Appl. (2023), 21:62 11 Iterations new iteration defined as in (3.1) iteration defined as in (2.3) iteration defined as in (2.2) I y1 = (0.607316, 0.607316) y1 = (0.715819, 0.715819) y1 = (0.715819, 0.715819) x2 = (0.566583, 0.566583) x2 = (0.597018, 0.597018) x2 = (0.631199, 0.631199) II y2 = (0.286736, 0.286736) y2 = (0.456532, 0.456532) y2 = (0.489716, 0.489716) x3 = (0.091520, 0.091520) x3 = (0.179742, 0.179742) x3 = (0.344357, 0.344357) III y3 = (0.045760, 0.045760) y3 = (0.092728, 0.092728) y3 = (0.191416, 0.191416) x4 = (0.000048, 0.000048) x4 = (0.002857, 0.002857) x4 = (0.172179, 0.172179) IV y4 = (0.000024, 0.000024) y4 = (0.001428, 0.001428) y4 = (0.086520, 0.086520) x5 = (0.000000, 0.000000) x5 = (0.000000, 0.000000) x5 = (0.086090, 0.086090) V y5 = (0.000000, 0.000000) y5 = (0.000000, 0.000000) y5 = (0.043047, 0.043047) x6 = (0.000000, 0.000000) x6 = (0.000000, 0.000000) x6 = (0.043045, 0.043045) V I y6 = (0.021522, 0.021522) x7 = (0.021522, 0.021522) V II y7 = (0.010761, 0.010761) x8 = (0.010761, 0.010761) V III y8 = (0.005381, 0.005381) x9 = (0.005381, 0.005381) IX y9 = (0.002690, 0.002690) x10 = (0.002690, 0.002690) X y10 = (0.001345, 0.001345) x11 = (0.001345, 0.001345) XI y11 = (0.000673, 0.000673) x12 = (0.000673, 0.000673) XII y12 = (0.000336, 0.000336) x13 = (0.000336, 0.000336) XIII y13 = (0.000168, 0.000168) x14 = (0.000168, 0.000168) XIV y14 = (0.000084, 0.000084) x15 = (0.000084, 0.000084) XV y15 = (0.000042, 0.000042) x16 = (0.000042, 0.000042) XV I y16 = (0.000021, 0.000021) x17 = (0.000021, 0.000021) XV II y17 = (0.000011, 0.000011) x18 = (0.000011, 0.000011) XV III y18 = (0.000005, 0.000005) x19 = (0.000005, 0.000005) XIX y19 = (0.000003, 0.000003) x20 = (0.000003, 0.000003) XX y20 = (0.000001, 0.000001) x21 = (0.000001, 0.000001) XXI y21 = (0.000001, 0.000001) x22 = (0.000001, 0.000001) XXII y22 = (0.000000, 0.000000) x23 = (0.000000, 0.000000) 12 Int. 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Leustean, Nonexpansive Iterations in Uniformly Convex w-Hyperbolic Spaces, In: Nonlinear Analysis and Opti- mization I: Nonlinear Analysis, vol. 513, pp. 193-209, 2010. [28] T.C. Lim, Remarks on Some Fixed Point Theorems, Proc. Amer. Math. Soc. 60 (1976), 179-182. [29] A.R. Khan, H. Fukhar-ud-din, M.A. Ahmad Khan, An Implicit Algorithm for Two Finite Families of Nonex- pansive Maps in Hyperbolic Spaces, Fixed Point Theory Appl. 2012 (2012), 54. https://doi.org/10.1186/ 1687-1812-2012-54. https://doi.org/10.1006/jath.1996.3093 https://doi.org/10.1515/dema-2016-0010 https://doi.org/10.2996/kmj/1138846111 https://doi.org/10.1017/CBO9780511526152 https://doi.org/10.1017/CBO9780511526152 https://doi.org/10.1186/1687-1812-2012-54 https://doi.org/10.1186/1687-1812-2012-54 1. Introduction 2. Preliminaries 3. Main Results References