EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 3, 2021, 650-665 ISSN 1307-5543 – ejpam.com Published by New York Business Global Mixed type algorithms for asymptotically nonexpansive mappings in hyperbolic spaces Tanakit Thianwan Department of Mathematics, School of Science, University of Phayao, Phayao, 56000, Thailand Abstract. In this paper, a new mixed type iteration process for approximating a common fixed point of two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings is constructed. We then establish a strong convergence theorem under mild conditions in a uniformly convex hyperbolic space. The results presented here extend and improve some related results in the literature. 2020 Mathematics Subject Classifications: 47H10, 47H09, 46B20 Key Words and Phrases: uniformly convex hyperbolic space, strong convergence, common fixed points, mixed type algorithm, asymptotically nonexpansive mapping 1. Introduction and preliminaries Iterative schemes play a prominent role in approximating fixed points of nonlinear mappings. Structural properties of the underlying space, such as strict convexity and uniform convexity, are very much needed for the development of iterative fixed point theory in it. Hyperbolic spaces are general in nature and inherit rich geometrical structure suitable to obtain new results in topology, graph theory, multi-valued analysis and metric fixed point theory. Fixed-point iteration processes for nonexpansive self and nonself mappings have been studied extensively by various authors to solve the nonlinear operator equations in Hilbert spaces and Banach spaces (see [10, 11, 17–19, 21, 28, 30, 33, 37] and the references cited therein). Goebel and Kirk [6], in 1972, introduced the class of asymptotically nonexpansive self-mappings, which is an important generalization of the class of nonexpansive self- mappings. In the last few decades investigations of fixed points by some iterative schemes for asymptotically nonexpansive mappings have attracted many mathematicians. In 1991, Schu [28] introduced the following modified Mann iteration process un+1 = (1− ϑn)un + ϑnT nun, n ≥ 1, (1) DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.4005 Email address: tanakit.th@up.ac.th (T. Thianwan) http://www.ejpam.com 650 © 2021 EJPAM All rights reserved. T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 651 to approximate fixed points of asymptotically nonexpansive self-mappings in a Hilbert space. Since then, Schu’s iteration process (1) has been widely used to approximate fixed points of asymptotically nonexpansive self-mappings in Hilbert spaces or Banach spaces ([4, 18, 22, 24, 28, 29, 34]). In 2003, Chidume, Ofoedu, and Zegeye [5] introduced the concept of asymptotically nonexpansive nonself-mappings. Also, they studied the following iterative sequence un+1 = P((1− ϑn)un + ϑnT (PT )n−1un) (2) to approximate some fixed point of T under suitable conditions. If T is a self-mapping, then P becomes the identity mapping so that (2) reduces to (1). In 2006, Wang [36] considered the following iteration process which is a generalization of (2), vn = P((1− ζn)un + ζnT2(PT 2)n−1un), un+1 = P((1− ϑn)un + ϑnT1(PT 1)n−1vn), n ≥ 1, (3) where T1, T2 : K → X are asymptotically nonexpansive nonself-mappings and {ϑn} and {ζn} are real sequences in [0,1). Meanwhile, the results of [36] generalized the results of [5]. The projection type Ishikawa iteration process for approximating common fixed points of two asymptotically nonexpansive nonself-mappings was defined and constructed by Thianwan [35] in a uniformly convex Banach space as follows: vn = P((1− ζn)un + ζnT2(PT 2)n−1un), un+1 = P((1− ϑn)vn + ϑnT1(PT 1)n−1vn), n ≥ 1, (4) where {ϑn} and {ζn} are appropriate real sequences in [0,1). Note that Thianwan process (4) and Wang process (3) are independent neither reduces to the other. In 2012, Guo, Cho and Guo [9] studied the following iteration scheme: vn = P((1− ζn)Sn2 un + ζnT2(PT 2)n−1un), un+1 = P((1− ϑn)Sn1 un + ϑnT1(PT 1)n−1vn), n ≥ 1, (5) where S1,S2 : K → K are asymptotically nonexpansive self-mappings, T1, T2 : K → X are asymptotically nonexpansive nonself-mappings and {ϑn}, {ζn} are two sequences in [0,1) to approximate common fixed points of S1,S2, T1 and T2 under proper conditions. The class of hyperbolic spaces, nonlinear in nature, is a general abstract theoretic set- ting with rich geometrical structure for metric fixed point theory. The study of hyperbolic spaces has been largely motivated and dominated by questions about hyperbolic groups, one of the main objects of study in geometric group theory. Fixed point theory and hence approximation techniques have been extended to hyperbolic spaces (see [1–3, 25–27] and references therein). T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 652 Throughout this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [13], defined below, which play a significant role in many branches of mathe- matics. A hyperbolic space (X , d,H) is a metric space (X , d) together with a mapping H : X × X × [0, 1]→ X satisfying (H1) : d(z,H(u, v, ψ)) ≤ (1− ψ) d(z, u) + ψd(z, v), (H2) : d(H(u, v, ψ)),H(u, v, µ) = |ψ − µ| d(u, v), (H3) : H(u, v, ψ) = H(v, u, 1− ψ), (H4) : d(H(u, z, ψ),H(v, w, ψ)) ≤ (1− ψ) d(u, v) + ψd(z, w) for all u, v, w, z ∈ X and ψ, µ ∈ [0, 1] . A subset K of a hyperbolic space X is convex if H(u, v, ψ) ∈ K for all u, v ∈ K and ψ ∈ [0, 1] . If a space satisfies only (H1), it coincides with the convex metric space introduced by Takahashi [32]. The concept of hyperbolic spaces in [13] is more restrictive than the hyperbolic type introduced by Goebel et al. [7] since (H1) − (H3) together are equivalent to (X , d,H) being a space of hyperbolic type in [7]. Also it is slightly more general than the hyperbolic space defined by Reich et al. [23]. A hyperbolic space (X , d,H) is said to be (i) strictly convex [32] if for any u, v ∈ X and ψ ∈ [0, 1] , there exists a unique element z ∈ X such that d(z, u) = ψd(u, v) and d(z, v) = (1− ψ)d(u, v); (ii) uniformly convex [31] if for all u, v, w ∈ X , r > 0 and ε ∈ (0, 2], there exists δ ∈ (0, 1] such that d(H(u, v, 1 2), u) ≤ (1− δ)r whenever d(u,w) ≤ r, d(v, w) ≤ r and d(u, v) ≥ εr. A mapping η : (0,∞) × (0, 2] → (0, 1] providing such δ = η(r, ε) for given r > 0 and ε ∈ (0, 2] is called modulus of uniform convexity. We call η monotone if it decreases with r (for a fixed ε). A uniformly convex hyperbolic space is strictly convex (see [15]). In the sequel, let (X , d) be a metric space, and let K be a nonempty subset of X . We shall denote the fixed point set of a mapping T by F(T ) = {u ∈ K : T u = u} and d(u,F(T )) = inf {d(u, p) : p ∈ F(T )} . A self-mapping T is said to be nonexpansive if d(T u, T v) ≤ d(u, v) for all u, v ∈ K. T : K → K is called asymptotically nonexpansive if there exists a sequence {kn} ⊂ [1,∞) with kn → 1 such that d(T nu, T nv) ≤ knd(u, v) (6) for all u, v ∈ K and n ≥ 1. T : K → K is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that d(T nu, T nv) ≤ Ld(u, v) for all u, v ∈ K and n ≥ 1. It follows that each nonexpansive mapping is an asymptotically nonexpansive mapping with kn = 1, ∀n ≥ 1. Moreover, each asymptotically nonexpansive mapping is a uniformly L -Lipschitzian mapping with L = supn∈ℵ {kn} . However, the converse of these statements is not true, in general. Note that, a subset K of X is said to be a retract if there exists a continuous mapping P : X → K such that Pu = u for all u ∈ K. For more information on nonexpansive retracts and retractions, we refer the reader to ([8, 14]). For any nonempty subset K of a real metric space (X , d), let P : X → K be a nonexpan- sive retraction of X onto K. Then, T : K → X is said to be an asymptotically nonexpansive T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 653 nonself-mapping (see [5]) if there exists a sequence {kn} ⊂ [1,∞) with kn → 1 as n→∞ such that d(T (PT )n−1 u, T (PT )n−1 v) ≤ knd (u, v) (7) for all u, v ∈ K and n ≥ 1. We denote by (PT )0 the identity map from K onto itself. We see that if T is a self-mapping, then P becomes the identity mapping, so that (7) reduces to (6). In addition, if T : K → X is asymtotically nonexpansive in light of (7) and P : X → K is a nonexpansive retraction, then PT : K → K is asymtotically nonexpansive in light of (6) (see also (8)). Indeed, for all u, v ∈ K and n ≥ 1, by (7), it follows that d((PT )nu, (PT )nv) = d(PT (PT )n−1u,PT (PT )n−1v) ≤ d(T (PT )n−1u, T (PT )n−1v) ≤ knd(u, v). Therefore, we now introduce the following definition. Definition 1. For any nonempty subset K of a metric space (X , d). Let P : X → K be a nonexpansive retraction of X onto K. A nonself-mapping T : K → X is called asymptotically nonexpansive with respect to P if there exists a sequence {kn} ⊂ [1,∞) with kn → 1 as n→∞ such that d((PT )n u, (PT )n v) ≤ knd (u, v) (8) for all u, v ∈ K and n ≥ 1. In the sequel, we shall need the following lemmas. Lemma 1. ([20]) Let {ηn} , {ϑn} and {ζn} be sequences of non-negative real numbers such that ηn+1 ≤ (1 + ϑn)ηn + ζn, ∀n ≥ 1. If ∞∑ n=1 ϑn <∞ and ∞∑ n=1 ζn <∞, then lim n→∞ ηn exists. Lemma 2. ([12]) Let {un} and {vn} be two sequences of a uniformly convex hyperbolic space (X , d, H) such that, for r ∈ [0,∞), lim n→∞ sup d(un, a) ≤ r, lim n→∞ sup d(vn, a) ≤ r, and lim n→∞ d(H(un, vn, ςn), a) = r, where ςn ∈ [a, b] with 0 < a ≤ b < 1, then we have, lim n→∞ d(un, vn) = 0. T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 654 2. Main results In this section, we suggest a new iterative algorithm for mixed type asypmtotically non- expansive mappings and establish the strong convergence theorem in a uniformly convex hyperbolic space. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space (X , d,H) and P : X → K be a nonexpansive retraction of X onto K. Let S1,S2 : K → K be two asymptotically nonexpasive self-mappings and T1, T2 : K → X be two asymptotically nonexpasive nonself-mappings. We will denote the set of common fixed points of S1, S2, T1 and T2 by F , that is, F := F(S1)∩F(S2)∩F(T1)∩F(T2). The algorithm is defined as follows: u1 ∈ K, vn = H(Sn2 un, (PT 2)nun, ζn), un+1 = H(Sn1 vn, (PT 1)nvn, ϑn), (9) where {ϑn} and {ζn} are two sequences in [0, 1). The following lemmas are needed. Lemma 3. Let (X , d,H) be a uniformly convex hyperbolic space and K be a nonempty closed convex subset of X . Let S1,S2 : K → K be two asymptotically nonexpasive self- mappings with {k(1) n }, {k(2) n } ⊂ [1,∞) and T1, T2 : K → X be two asymptotically non- expasive nonself-mappings with {l(1) n }, {l(2) n } ⊂ [1,∞) such that ∞∑ n=1 (k(i) n − 1) < ∞ and ∞∑ n=1 (l(i)n −1) <∞ for i = 1, 2, respectively and F 6= ∅. Suppose that {ϑn} and {ζn} are real sequences in [0, 1). From an arbitrary u1 ∈ K, define the sequence {un} using algorithm (9). Then lim n→∞ d (un, q) exists for any q ∈ F . Proof. Let q ∈ F . Setting hn = max{k(1) n , k (2) n , l (1) n , l (2) n }. Using algorithm (9), we have d (vn, q) = d (H (Sn2 un, (PT 2)n un, ζn) , q) ≤ (1− ζn) d (Sn2 un, q) + ζnd ((PT 2)n un, q) ≤ (1− ζn)hnd (un, q) + ζnhnd (un, q) = hnd (un, q) , (10) and so d (un+1, q) = d (H (Sn1 vn, (PT 1)n vn, ϑn) , q) ≤ (1− ϑn) d (Sn1 vn, q) + ϑnd ((PT 1)n vn, q) ≤ (1− ϑn)hnd (vn, q) + ϑnhnd (vn, q) = hnd (vn, q) ≤ h2 nd (un, q) T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 655 = ( 1 + (h2 n − 1) ) d (un, q) . (11) Since ∞∑ n=1 (k(i) n − 1) < ∞ and ∞∑ n=1 (l(i)n − 1) < ∞ for i = 1, 2, we have ∞∑ n=1 (h2 n − 1) < ∞. It follows from Lemma 1 that limn→∞ d (un, q) exists. Lemma 4. Let (X , d,W) be a uniformly convex hyperbolic space and K be a nonempty closed convex subset of X . Let S1,S2 : K → K be two asymptotically nonexpansive self- mappings with {k(1) n }, {k(2) n } ⊂ [1,∞) and T1, T2 : K → X be two asymptotically non- expansive nonself-mappings with {l(1) n }, {l(2) n } ⊂ [1,∞) such that ∞∑ n=1 (k(i) n − 1) < ∞ and ∞∑ n=1 (l(i)n − 1) < ∞ for i = 1, 2, respectively, and F 6= ∅. From an arbitrary u1 ∈ K, define the sequence {un} using algorithm (9) and the following conditions hold: (i) {ϑn} and {ζn} are real sequences in [ε, 1− ε] for some ε ∈ (0, 1); (ii) d(u, Tiv) ≤ d(Siu, Tiv) for all u, v ∈ K and i = 1, 2. Then, lim n→∞ d(un,Siun) = lim n→∞ d(un, (PT i)un) = 0 for i = 1, 2. Proof. Let q ∈ F . Set hn = max{k(1) n , k (2) n , l (1) n , l (2) n }. By Lemma 3, we have lim n→∞ d(un, q) exists. Assume that lim n→∞ d(un, q) = c, letting n→∞ in the inequality (11), we have lim n→∞ d(H(Sn1 vn, (PT 1)nvn, ϑn), q) = c. (12) In addition, using (10), we have d(Sn1 vn, q) ≤ h2 nd(un, q). Taking the lim sup on both sides in this inequality, we have lim sup n→∞ d(Sn1 vn, q) ≤ c. (13) Taking the lim sup on both sides in the inequality (10), we obtain lim sup n→∞ d(vn, q) ≤ c, and so lim sup n→∞ d((PT 1)nvn, q) ≤ lim sup n→∞ hnd(vn, q) = c. (14) Using (12), (13), (14), and Lemma 2, we have lim n→∞ d(Sn1 vn, (PT 1)nvn) = 0. (15) By the condition (ii), we have T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 656 d(vn, (PT 1)nvn) ≤ d(Sn1 vn, (PT 1)nvn). (16) Letting n→∞ in the inequality (16), by (15), we have lim n→∞ d(vn, (PT 1)nvn) = 0. (17) Using (11), we have d(un+1, q) ≤ (1− ϑn)d(Sn1 vn, q) + ϑnd((PT 1)nvn, q) ≤ (1− ϑn)d(Sn1 vn, q) + ϑnd(Sn1 vn, (PT 1)nvn) + ϑnd(Sn1 vn, q) = d(Sn1 vn, q) + ϑnd(Sn1 vn, (PT 1)nvn) ≤ hnd(vn, q) + ϑnd(Sn1 vn, (PT 1)nvn). (18) Taking the lim inf on both sides in the inequality (18), using (15), ∞∑ n=1 (hn − 1) < ∞ and lim n→∞ d(un+1, q) = c, we have lim inf n→∞ d(vn, q) ≥ c. (19) Since lim sup n→∞ d(vn, q) ≤ c, by (19), we have lim n→∞ d(vn, q) = c. This implies that c = lim n→∞ d(vn, q) ≤ lim n→∞ d(H(Sn2 un, (PT 2)nun, ζn), q) ≤ lim n→∞ d(un, q) = c, and so lim n→∞ d(H(Sn2 un, (PT 2)nun, ζn), q) = c. (20) In addition, we have lim sup n→∞ d(Sn2 un, q) ≤ lim sup n→∞ hnd(un, q) = c (21) and lim sup n→∞ d((PT 2)nun, q) ≤ lim sup n→∞ hnd(un, q) = c. (22) It follows from (20), (21), (22), and Lemma 2 that lim n→∞ d(Sn2 un, (PT 2)nun) = 0. (23) T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 657 Now, we prove that lim n→∞ d(un, (PT 1)un) = 0 = lim n→∞ d(un, (PT 2)un). Indeed, by the condition (ii), we have d(un, (PT 2)nun) ≤ d(Sn2 un, (PT 2)nun). (24) Using (23) and (24), we have lim n→∞ d(un, (PT 2)nun) = 0. (25) Using algorithm (9), we have d(vn,Sn2 un) ≤ (1− ζn)d(Sn2 un,Sn2 un) + ζnd(Sn2 un, (PT 2)nun) = ζnd(Sn2 un, (PT 2)nun). It follows from (23) that lim n→∞ d(vn,Sn2 un) = 0. (26) Furthermore, we have d(vn, un) ≤ d(vn,Sn2 un) + d(Sn2 un, (PT 2)nun) + d((PT 2)nun, un). (27) It follows from (23), (25), (26), and (27) that lim n→∞ d(un, vn) = 0. (28) By the condition (ii), we have d(un, (PT 1)nun) ≤ d(Sn1 un, (PT 1)nun). Since d(Sn1 un, (PT 1)nun) ≤ d(Sn1 un,Sn1 vn) + d(Sn1 vn, (PT 1)nvn) + d((PT 1)nvn, (PT 1)nun) ≤ hnd(un, vn) + d(Sn1 vn, (PT 1)nvn) + hnd(vn, un), (29) using (15), (28) and (29), we have lim n→∞ d(Sn1 un, (PT 1)nun) = 0, (30) T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 658 and so lim n→∞ d(un, (PT 1)nun) = 0. (31) In addition, d(un+1,Sn1 vn) = d((H(Sn1 vn, (PT 1)nvn, αn)),Sn1 vn) ≤ (1− ϑn)d(Sn1 vn,Sn1 vn) + ϑnd((PT 1)nvn,Sn1 vn) = ϑnd((PT 1)nvn,Sn1 vn). Thus, it follows from (15) that lim n→∞ d(un+1,Sn1 vn) = 0. (32) In addition d(un+1, (PT 1)nvn) ≤ d(un+1,Sn1 vn) + d(Sn1 vn, (PT 1)nvn). Using (15) and (32), we have lim n→∞ d(un+1, (PT 1)nvn) = 0. (33) It follows from (30) and (31) that d(Sn1 un, un) ≤ d(Sn1 un, (PT 1)nun) + d((PT 1)nun, un) → 0 (as n→∞). (34) In addition, d(Sn1 un, (PT 2)nun) ≤ d(Sn1 un, un) + d(un, (PT 2)nun). Thus, it follows from (25) and (34) that lim n→∞ d(Sn1 un, (PT 2)nun) = 0. (35) In addition, d(Sn1 vn, (PT 2)nun) ≤ d(Sn1 vn,Sn1 un) + d(Sn1 un, (PT 2)nun) ≤ hnd(vn, un) + d(Sn1 un, (PT 2)nun). Using (28) and (35), we have T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 659 lim n→∞ d(Sn1 vn, (PT 2)nun) = 0. (36) It follows from (28), (32) and (36) that d(un+1, (PT 2)nvn) ≤ d(un+1,Sn1 vn) + d(Sn1 vn, (PT 2)nun) + d((PT 2)nun, (PT 2)nvn) ≤ d(un+1,Sn1 vn) + d(Sn1 vn, (PT 2)nun) + hnd(un, vn) → 0 (as n→∞). (37) Again, since (PT i)(PT i) n−1vn−1, un ∈ K for i = 1, 2 and T1, T2 are two asymptotically nonexpansive nonself-mappings, we have d((PT i) nvn−1, (PT i)un) = d(((PT i)(PT i) n−1vn−1), (PT i)un) ≤ max{l(1) 1 , l (2) 1 }d((PT i) n−1vn−1, un). (38) Using (33), (37), and (38), for i = 1, 2, we have lim n→∞ d((PT i) nvn−1, (PT i)un) = 0. (39) Moreover, we have d(un+1, vn) ≤ d(un+1, (PT 1)nvn) + d((PT 1)nvn, vn). Using (17) and (33), we have lim n→∞ d(un+1, vn) = 0. (40) In addition, for i = 1, 2, we have d(un, (PT i)un) ≤ d(un, (PT i) nun) + d((PT i) nun, (PT i) nvn−1) + d((PT i) nvn−1, (PT i)un) ≤ d(un, (PT i) nun) +max{supn≥1l (1) n , supn≥1l (2) n }d(un, vn−1) + d((PT i) nvn−1, (PT i)un). Thus, it follows from (25), (31), (39), and (40) that lim n→∞ d(un, (PT 1)un) = lim n→∞ d(un, (PT 2)un) = 0. Finally, we prove that T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 660 lim n→∞ d(un,S1un) = lim n→∞ d(un,S2un) = 0. In fact, for i = 1, 2, we have d(un,Siun) ≤ d(un, (PT i) nun) + d(Siun, (PT i) nun) ≤ d(un, (PT i) nun) + d(Sni un, (PT i) nun). Thus, it follows from (23), (25), (30), and (31) that lim n→∞ d(un,S1un) = lim n→∞ d(un,S2un) = 0. The proof is completed. The following example presents the condition (ii) in Lemma 4 which is satisfied by the mappings Si and Ti, i = 1, 2, when S1 = S2 = S and T1 = T2 = T , where S and T are given in the next example. Example 1. ([16]) Let X be the real line with metric d(u, v) = |u − v| and K = [−1, 1]. Define H : X × X × [0, 1] → X by H(u, v, ψ) := ψu + (1 − ψ)v for all u, v ∈ X and ψ ∈ [0, 1]. Then (X , d,H) is a complete uniformly hyperbolic space with a monotone mod- ulus of uniform convexity and K is a nonempty closed convex subset of X . Define two mappings S, T : K → K by T u = { −2 sin u 2 , if u ∈ [0, 1], 2 sin u 2 , if u ∈ [−1, 0) and Su = { u, if u ∈ [0, 1], −u, if u ∈ [−1, 0). Clearly, F(T ) = {0} and F(S) = {u ∈ K; 0 ≤ u ≤ 1} . Now, we show that T is nonexpan- sive. In fact, if u, v ∈ [0, 1] or u, v ∈ [−1, 0), then d(T u, T v) = |T u− T v| = 2| sin u 2 − sin v 2 | ≤ |u− v| = d(u, v). If u ∈ [0, 1] and v ∈ [−1, 0) or u ∈ [−1, 0) and v ∈ [0, 1], then d(T u, T v) = |T u− T v| = 2| sin u 2 + sin v 2 | = 4| sin u+ v 4 cos u− v 4 | ≤ |u+ v| ≤ |u− v| T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 661 = d(u, v). That is, T is nonexpansive. It follows that T is an asymptotically nonexpansive mapping with kn = 1 for each n ≥ 1. Similarly, we can show that S is an asymptotically nonexpan- sive mapping with ln = 1 for each n ≥ 1. Next, to show that S and T satisfy the condition (ii) in Lemma 4, we have to consider the following cases: Case 1. Let u, v ∈ [0, 1]. It follows that d(u, T v) = |u− T v| = |u+ 2 sin v 2 | = |Su− T v| = d(Su, T v). Case 2. Let u, v ∈ [−1, 0). It follows that d(u, T v) = |u− T v| = |u− 2 sin v 2 | ≤ | − u− 2 sin v 2 | = |Su− T v| = d(Su, T v). Case 3. Let u ∈ [−1, 0) and v ∈ [0, 1]. It follows that d(u, T v) = |u− T v| = |u+ 2 sin v 2 | ≤ | − u+ 2 sin v 2 | = |Su− T v| = d(Su, T v). Case 4. Let u ∈ [0, 1] and v ∈ [−1, 0]. It follows that d(u, T v) = |u− T v| = |u− 2 sin v 2 | = |Su− T v| = d(Su, T v). Hence the condition (ii) in Lemma 4 is satisfied. Now, we can prove a strong convergence theorem. Theorem 1. Let K,X ,S1,S2, T1 and T2 satisfy the hypotheses of Lemma 4. Suppose that {ϑn}, {ζn} are real sequences in [ε, 1−ε] for some ε ∈ (0, 1) and Si, Ti for all i = 1, 2 satisfy the condition (ii) in Lemma 4. If there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0 and f(r) > 0 for all r ∈ (0,∞) such that f(d(u,F)) ≤ d(u,S1u) + d(u,S2u) + d(u, (PT 1)u) + d(u, (PT 2)u) for all u ∈ K, where d(u,F) = inf{d(u, q) : q ∈ F}. Then the sequence {un} defined by algorithm (9) converges strongly to a common fixed point of S1,S2, T1 and T2. Proof. By Lemma 4, we have lim n→∞ d(un,Siun) = lim n→∞ d(un, (PT i)un) = 0 for i = 1, 2. It follows from hypothesis that lim n→∞ f(d(un,F)) ≤ lim n→∞ (d(un,S1un) + d(un,S2un) + d(un, (PT 1)un) + d(un, (PT 2)un)) = 0. Thus lim n→∞ f(d(un,F)) = 0. Since f : [0,∞)→ [0,∞) is a nondecreasing function satisfying f(0) = 0, f(r) > 0 for all r ∈ (0,∞). Using Lemma 11, we have lim n→∞ d(un,F) exists. It implies that lim n→∞ d(un,F) = 0. Now, we show that {un} is a Cauchy sequence in K. In fact, from (11), we have d(un+1, q) ≤ (1 + (h2 n − 1))d(un, q) for each n ≥ 1, where hn = max{k(1) n , k (2) n , l (1) n , l (2) n } and q ∈ F . For any m,n,m > n ≥ 1, we have d(um, q) ≤ (1 + (h2 m−1 − 1))d(um−1, q) T. Thianwan / Eur. J. Pure Appl. Math, 14 (3) (2021), 650-665 662 ≤ eh 2 m−1−1d(um−1, q) ≤ eh 2 m−1−1e h2 m−2−1d(um−2, q) ... ≤ e ∑m−1 i=n (h2 i−1)d(un, q) ≤Md(un, q), where M = eΣ∞i=1(h2 i−1). Thus, for any q ∈ F , we have d(un, um) ≤ d(un, q) + d(um, q) ≤ (1 +M)d(un, q). Taking the infimum over all q ∈ F , we have d(un, um) ≤ (1 +M)d(un,F). Thus it follows from lim n→∞ d(un,F) = 0 that {un} is a Cauchy sequence. Since K is a closed subset in a complete hyperbolic space X , the sequence {un} converges strongly to some q∗ ∈ K. It is easy to prove that F(S1),F(S2),F(T1) and F(T2) are all closed and so F is a closed subset of K. Since lim n→∞ d(un,F) = 0 gives that d(q∗,F) = 0. Therefore q∗ ∈ F . This completes the proof. If T1 and T2 are self-mappings, then P becomes the identity mapping. By using to the same ideas and techniques as in Lemma 3, Lemma 4 and Theorem 1, we can also obtain a strong convergence theorem for asymptotically nonexpansive mappings in a uniformly convex hyperbolic space. Therefore we can state the following result without proofs. Theorem 2. Let (X , d,W) be a uniformly convex hyperbolic space and K be a nonempty closed convex subset of X . Let S1,S2 : K → K be two asymptotically nonexpansive map- pings with {k(1) n }, {k(2) n } ⊂ [1,∞) and T1, T2 : K → K be two asymptotically nonexpansive mappings with {l(1) n }, {l(2) n } ⊂ [1,∞) such that ∞∑ n=1 (k(i) n −1) <∞ and ∞∑ n=1 (l(i)n −1) <∞ for i = 1, 2, respectively, and F 6= ∅. Suppose that {ϑn}, {ζn} are real sequences in [ε, 1 − ε] for some ε ∈ (0, 1) and Si, Ti for all i = 1, 2 satisfy the condition (ii) in Lemma 4. If there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0 and f(r) > 0 for all r ∈ (0,∞) such that f(d(u,F)) ≤ d(u,S1u) + d(u,S2u) + d(u, T1u) + d(u, T2u) for all u ∈ K, where d(u,F) = inf{d(u, q) : q ∈ F}. Then the sequence {un} defined by vn = H(Sn2 un, T n 2 un, ζn), un+1 = H(Sn1 vn, T n 1 vn, ϑn) converges strongly to a common fixed point of S1,S2, T1 and T2. REFERENCES 663 3. Conclusions Author constructed a new mixed type iterative method to approximate a common fixed point for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings in a uniformly convex hyperbolic space. An asymptotically nonexpansive nonself-mapping with respect to a nonexpansive retraction is defined in Definition 1. An illustrative example is also provided as Example 1. Author proved strong convergence result which is stronger than that of delta and weak convergence results. 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