Convergence Theorems for Total Asymptotically Nonexpansive Mappings in Hyperbolic Spaces

Abstract

The purpose of this paper is to introduce the concept of total asymptotically nonexpansive mappings and to prove some Δ-convergence theorems of the iteration process for this kind of mappings in the setting of hyperbolic spaces. The results presented in the paper extend and improve some recent results announced in the current literature.

1. Introduction and Preliminaries

Most of the problems in various disciplines of science are nonlinear in nature whereas fixed point theory proposed in the setting of normed linear spaces or Banach spaces majorly depends on the linear structure of the underlying spaces. A nonlinear framework for fixed point theory is a metric space embedded with a “convex structure.” The class of hyperbolic spaces, nonlinear in nature, is a general abstract theoretic setting 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.

Throughout this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [1], defined below, which is more restrictive than the hyperbolic type introduced in [2] and more general than the concept of hyperbolic space in  [3].

A hyperbolic space is a metric space (𝑋,𝑑) together with a mapping 𝑊∶𝑋²×[0,1]→𝑋 satisfying (i)𝑑(𝑢,𝑊(𝑥,𝑦,𝛼))≤(1−𝛼)𝑑(𝑢,𝑥)+𝛼𝑑(𝑢,𝑦); (ii)𝑑(𝑊(𝑥,𝑦,𝛼),𝑊(𝑥,𝑦,𝛽))=|𝛼−𝛽|𝑑(𝑥,𝑦); (iii)𝑊(𝑥,𝑦,𝛼)=𝑊(𝑦,𝑥,(1−𝛼)); (iv)𝑑(𝑊(𝑥,𝑧,𝛼),𝑊(𝑦,𝑤,𝛼))≤(1−𝛼)𝑑(𝑥,𝑦)+𝛼𝑑(𝑧,𝑤),for all 𝑥,𝑦,𝑧,𝑤∈𝑋 and 𝛼,𝛽∈[0,1]. A nonempty subset 𝐾 of a hyperbolic space 𝑋 is convex if 𝑊(𝑥,𝑦,𝛼)∈𝐾 for all 𝑥,𝑦∈𝐾 and 𝛼∈[0,1]. The class of hyperbolic spaces contains normed spaces and convex subsets thereof, the Hilbert ball equipped with the hyperbolic metric [4], Hadamard manifolds, and CAT(0) spaces in the sense of Gromov (see [5]).

A hyperbolic space is uniformly convex [6] if for any 𝑟>0 and 𝜖∈(0,2] there exists a 𝛿∈(0,1], such that, for all 𝑢,𝑥,𝑦∈𝑋, we have 𝑑𝑊1𝑥,𝑦,2,𝑢≤(1−𝛿)𝑟, (1) provided 𝑑(𝑥,𝑢)≤𝑟, 𝑑(𝑦,𝑢)≤𝑟 and 𝑑(𝑥,𝑦)≥𝜖𝑟.

A map 𝜂∶(0,∞)×(0,2]→(0,1], which provides such a 𝛿=𝜂(𝑟,𝜖) for given 𝑟>0 and 𝜖∈(0,2], is known as a modulus of uniform convexity of 𝑋. We call 𝜂 monotone if it decreases with 𝑟 (for a fixed 𝜖), that is, for all 𝜖>0, for all 𝑟₂≥𝑟₁>0(𝜂(𝑟₂,𝜖)≤𝜂(𝑟₁,𝜖)).

In the sequel, let (𝑋,𝑑) be a metric space and let 𝐾 be a nonempty subset of 𝑋. We will denote the fixed point set of a mapping 𝑇 by 𝐹(𝑇)={𝑥∈𝐾∶𝑇𝑥=𝑥}.

A mapping 𝑇∶𝐾→𝐾 is said to be nonexpansive, if 𝑑𝑇𝑥,𝑇𝑦≤𝑑𝑥,𝑦,∀𝑥,𝑦∈𝐾. (2)

A mapping 𝑇∶𝐾→𝐾 is said to be asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that 𝑑𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝑘_𝑛𝑑𝑥,𝑦,∀𝑛≥1,𝑥,𝑦∈𝐾. (3)

A mapping 𝑇∶𝐾→𝐾 is said to be uniformly L-Lipschitzian if there exists a constant 𝐿>0 such that 𝑑𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝐿𝑑𝑥,𝑦,∀𝑛≥1,𝑥,𝑦∈𝐾. (4)

Definition 1. A mapping 𝑇∶𝐾→𝐾 is said to be ({𝜇_𝑛},{𝜉_𝑛},𝜙)-total asymptotically nonexpansive, if there exist nonnegative sequences {𝜇_𝑛},{𝜉_𝑛} with 𝜇_𝑛→0,  𝜉_𝑛→0 and a strictly increasing continuous function 𝜙∶[0,∞)→[0,∞) with 𝜙(0)=0 such that 𝑑𝑇^𝑛𝑥,𝑇^𝑛𝑦≤𝑑𝑥,𝑦+𝜇_𝑛𝜙𝑑𝑥,𝑦+𝜉_𝑛,∀𝑛≥1,𝑥,𝑦∈𝐾. (5)

Remark 2. From the definitions, it is clear that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence {𝑘_𝑛=1} and each asymptotically nonexpansive mapping is a ({𝜇_𝑛},{𝜉_𝑛},𝜙)-total asymptotically nonexpansive mapping with 𝜉_𝑛=0, 𝜇_𝑛=𝑘_𝑛−1,𝑛≥1 and 𝜙(𝑡)=𝑡,𝑡≥0.

The existence of fixed points of various nonlinear mappings has relevant applications in many branches of nonlinear analysis and topology. On the other hand, there are certain situations where it is difficult to derive conditions for the existence of fixed points for certain types of nonlinear mappings. It is worth mentioning that fixed point theory for nonexpansive mappings, a limit case of a contraction mapping when the Lipschitz constant is allowed to be 1, requires tools far beyond from metric fixed point theory. Iteration schemas are the only main tool for analysis of generalized nonexpansive mappings. Fixed point theory has a computational flavor as one can define effective iteration schemas for the computation of fixed points of various nonlinear mappings. The problem of finding a common fixed point of some nonlinear mappings acting on a nonempty convex domain often arises in applied mathematics.

The purpose of this paper is to introduce the concept of total asymptotically nonexpansive mappings and to prove some Δ-convergence theorems of the iteration process for the approximation of total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results given in [6–18].

In order to define the concept Δ-convergence in the general setup of hyperbolic spaces, we first collect some basic concepts.

Let {𝑥_𝑛} be a bounded sequence in a hyperbolic space 𝑋. For 𝑥∈𝑋, we define a continuous functional 𝑟(⋅,{𝑥_𝑛})∶𝑋→[0,∞) by 𝑟𝑥,𝑥_𝑛=limsup_𝑛→∞𝑑𝑥,𝑥_𝑛. (6)

The asymptotic radius  𝑟({𝑥_𝑛}) of {𝑥_𝑛} is given by 𝑟𝑥_𝑛=inf𝑟𝑥,𝑥_𝑛∶𝑥∈𝑋. (7)

The asymptotic center 𝐴_𝑘({𝑥_𝑛}) of a bounded sequence {𝑥_𝑛} with respect to 𝐾⊂𝑋 is the set 𝐴_𝐾𝑥_𝑛=𝑥∈𝑋∶𝑟𝑥,𝑥_𝑛≤𝑟𝑦,𝑥_𝑛,∀𝑦∈𝐾. (8)

This is the set of minimizers of the functional 𝑟(⋅,{𝑥_𝑛}). If the asymptotic center is taken with respect to 𝑋, then it is simply denoted by 𝐴({𝑥_𝑛}). It is known that uniformly convex Banach spaces and CAT(0) spaces enjoy the property that “bounded sequences have unique asymptotic centers with respect to closed convex subsets.” The following lemma is due to Leuştean [19] and ensures that this property also holds in a complete uniformly convex hyperbolic space.

Lemma 3 (see [19]). Let (𝑋,𝑑,𝑊) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. Then, every bounded sequence {𝑥_𝑛} in 𝑋 has a unique asymptotic center with respect to any nonempty closed convex subset 𝐾 of 𝑋.
Recall that a sequence {𝑥_𝑛} in 𝑋 is said to Δ-converge to 𝑥∈𝑋 if 𝑥 is the unique asymptotic center of {𝑢_𝑛} for every subsequence {𝑢_𝑛} of {𝑥_𝑛}. In this case, we write Δ-lim_𝑛→∞𝑥_𝑛=𝑥 and call 𝑥 the Δ-𝑙𝑖𝑚𝑖𝑡 of {𝑥_𝑛}.
A mapping 𝑇∶𝐾→𝐾 is semicompact if every bounded sequence {𝑥_𝑛}⊂𝐾, satisfying 𝑑(𝑥_𝑛,𝑇𝑥_𝑛)→0, has a convergent subsequence.

Lemma 4 (see [8]). Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝛿_𝑛} be sequences of nonnegative real numbers satisfying 𝑎_𝑛+1≤1+𝛿_𝑛𝑎_𝑛+𝑏_𝑛,∀𝑛≥1. (9) If ∑^∞_𝑛=1𝛿_𝑛<∞ and ∑^∞_𝑛=1𝑏_𝑛<∞, then the limit lim_𝑛→∞𝑎_𝑛 exists. If there exists a subsequence {𝑎_𝑛𝑖}⊂{𝑎_𝑛} such that 𝑎_𝑛𝑖→0, then lim_𝑛→∞𝑎_𝑛=0.

Lemma 5 (see [11]). Let (𝑋,𝑑,𝑊) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity 𝜂. Let 𝑥∈𝑋 and {𝛼_𝑛} be a sequence in [𝑎,𝑏] for some 𝑎,𝑏∈(0,1). If {𝑥_𝑛} and {𝑦_𝑛} are sequences in 𝑋 such that limsup_𝑛→∞𝑑𝑥_𝑛,𝑥≤𝑐,limsup_𝑛→∞𝑑𝑦_𝑛,𝑥≤𝑐,lim_𝑛→∞𝑑𝑊𝑥_𝑛,𝑦_𝑛,𝛼_𝑛,𝑥=𝑐, (10) for some 𝑐≥0, then lim_𝑛→∞𝑑(𝑥_𝑛,𝑦_𝑛)=0.

Lemma 6 (see [11]). Let 𝐾 be a nonempty closed convex subset of uniformly convex hyperbolic space and {𝑥_𝑛} a bounded sequence in 𝐾 such that 𝐴({𝑥_𝑛})={𝑦} and 𝑟({𝑥_𝑛})=𝜌. If {𝑦_𝑚} is another sequence in 𝐾 such that lim_𝑚→∞𝑟(𝑦_𝑚,{𝑥_𝑛})=𝜌, then lim_𝑚→∞𝑦_𝑚=𝑦.

2. Main Results

Theorem 7. Let 𝐾 be a nonempty closed convex subset of a complete uniformly convex hyperbolic space 𝑋 with monotone modulus of uniform convexity 𝜂. Let 𝑇_𝑖∶𝐾→𝐾,𝑖=1,2, be uniformly 𝐿_𝑖-Lipschitzian and ({𝜇_𝑖𝑛},{𝜉_𝑖𝑛},𝜙_𝑖)-total asymptotically nonexpansive mappings with sequences {𝜇_𝑖𝑛} and {𝜉_𝑖𝑛} satisfying lim_𝑛→∞𝜇_𝑖𝑛=0, lim_𝑛→∞𝜉_𝑖𝑛=0, and strictly increasing function 𝜙_𝑖∶[0,∞)→[0,∞) with 𝜙_𝑖(0)=0,  𝑖=1,2. Assume that 𝐹∶=⋂²_𝑖=1𝐹(𝑇_𝑖)≠∅, and for arbitrarily chosen 𝑥₁∈𝐾, {𝑥_𝑛} is defined as follows: 𝑥_𝑛+1=𝑊𝑥_𝑛,𝑇^𝑛₁𝑦_𝑛,𝛼_𝑛,𝑦_𝑛=𝑊𝑥_𝑛,𝑇^𝑛₂𝑥_𝑛,𝛽_𝑛, (11) where {𝜇_𝑖𝑛}, {𝜉_𝑖𝑛}, 𝜙_𝑖, 𝑖=1,2, {𝛼_𝑛}, and {𝛽_𝑛} satisfy the following conditions: (1)∑^∞_𝑛=1𝜇_𝑖𝑛<∞,∑^∞_𝑛=1𝜉_𝑖𝑛<∞,𝑖=1,2; (2)there exist constants 𝑎,𝑏∈(0,1) with 0<𝑏(1−𝑎)≤1/2 such that {𝛼_𝑛}⊂[𝑎,𝑏] and {𝛽_𝑛}⊂[𝑎,𝑏]; (3)there exist a constant 𝑀^∗>0 such that 𝜙_𝑖(𝑟)≤𝑀^∗𝑟,𝑟>0,𝑖=1,2. Then, the sequence {𝑥_𝑛} defined by (11) Δ-converges to a common fixed point of 𝐹∶=⋂²_𝑖=1𝐹(𝑇_𝑖).

Proof. The proof of Theorem 7 is divided into four steps.
Step 1. First, we prove that lim_𝑛→∞𝑑(𝑥_𝑛,𝑝) exists for each 𝑝∈𝐹.
Set 𝜇_𝑛=max{𝜇_1𝑛,𝜇_2𝑛}, 𝜉_𝑛=max{𝜉_1𝑛,𝜉_2𝑛}, and 𝐿=max{𝐿₁,𝐿₂}. Since ∑^∞_𝑛=1𝜇_𝑖𝑛<∞,∑^∞_𝑛=1𝜉_𝑖𝑛<∞,  𝑖=1,2, ∑^∞_𝑛=1𝜇_𝑛<∞,  ∑^∞_𝑛=1𝜉_𝑛<∞. For any 𝑝∈𝐹(𝑇₁)∩𝐹(𝑇₂), by (11) we have 𝑑𝑥_𝑛+1𝑊𝑥,𝑝=𝑑_𝑛,𝑇^𝑛₁𝑦_𝑛,𝛼_𝑛≤,𝑝1−𝛼_𝑛𝑑𝑥_𝑛,𝑝+𝛼_𝑛𝑑𝑇^𝑛₁𝑦_𝑛≤,𝑝1−𝛼_𝑛𝑑𝑥_𝑛,𝑝+𝛼_𝑛𝑑𝑦_𝑛,𝑝+𝜇_𝑛𝜙₁𝑑𝑦_𝑛,𝑝+𝜉_𝑛≤1−𝛼_𝑛𝑑𝑥_𝑛,𝑝+𝛼_𝑛1+𝜇_𝑛𝑀^∗𝑑𝑦_𝑛,𝑝+𝜉_𝑛, (12) where 𝑑𝑦_𝑛𝑊𝑥,𝑝=𝑑_𝑛,𝑇^𝑛₂𝑥_𝑛,𝛽_𝑛≤,𝑝1−𝛽_𝑛𝑑𝑥_𝑛,𝑝+𝛽_𝑛𝑑𝑇^𝑛₂𝑥_𝑛≤,𝑝1−𝛽_𝑛𝑑𝑥_𝑛,𝑝+𝛽_𝑛𝑑𝑥_𝑛,𝑝+𝜇_𝑛𝜙₂𝑑𝑥_𝑛,𝑝+𝜉_𝑛≤1−𝛽_𝑛𝑑𝑥_𝑛,𝑝+𝛽_𝑛1+𝜇_𝑛𝑀^∗𝑑𝑥_𝑛,𝑝+𝜉_𝑛≤1+𝛽_𝑛𝜇_𝑛𝑀^∗𝑑𝑥_𝑛,𝑝+𝛽_𝑛𝜉_𝑛. (13) Substituting (13) into (12), we have 𝑑𝑥_𝑛+1≤,𝑝1−𝛼_𝑛𝑑𝑥_𝑛,𝑝+𝛼_𝑛1+𝜇_𝑛𝑀^∗1+𝛽_𝑛𝜇_𝑛𝑀^∗𝑑𝑥_𝑛,𝑝+𝛽_𝑛𝜉_𝑛+𝜉_𝑛≤1+1+𝛽_𝑛+𝛽_𝑛𝜇_𝑛𝑀^∗𝛼_𝑛𝜇_𝑛𝑀^∗𝑑𝑥_𝑛+,𝑝1+𝛽_𝑛+𝛽_𝑛𝜇_𝑛𝑀^∗𝛼_𝑛𝜉_𝑛. (14) Applying Lemma 4 to the inequality, we get that lim_𝑛→∞𝑑(𝑥_𝑛,𝑝) exist for 𝑝∈𝐹(𝑇₁)∩𝐹(𝑇₂).
Step  2. We show that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇₁𝑥_𝑛)=lim_𝑛→∞𝑑(𝑥_𝑛,𝑇₂𝑥_𝑛)=0.
For each 𝑝∈𝐹(𝑇₁)∩𝐹(𝑇₂), from the proof of Step 1, we know that lim_𝑛→∞𝑑(𝑥_𝑛,𝑝) exists. We may assume that lim_𝑛→∞𝑑(𝑥_𝑛,𝑝)=𝑐≥0. The case 𝑐=0 is trivial. Next, we deal with the case 𝑐>0. From (13), we have 𝑑𝑦_𝑛,𝑝≤1+𝛽_𝑛𝜇_𝑛𝑀^∗𝑑𝑥_𝑛,𝑝+𝛽_𝑛𝜉_𝑛. (15) Taking limsup on both sides in (15), we have limsup_𝑛→∞𝑑𝑦_𝑛,𝑝≤𝑐. (16) In addition, since 𝑑𝑇^𝑛₁𝑦_𝑛,𝑝≤𝑑𝑦_𝑛,𝑝+𝜇_𝑛𝜙₁𝑑𝑦_𝑛,𝑝+𝜉_𝑛≤1+𝜇_𝑛𝑀^∗𝑑𝑦_𝑛,𝑝+𝜉_𝑛, (17) we have limsup_𝑛→∞𝑑𝑇^𝑛₁𝑦_𝑛,𝑝≤𝑐. (18) Since lim_𝑛→∞𝑑(𝑥_𝑛+1,𝑝)=𝑐, it is easy to prove that lim_𝑛→∞𝑑𝑊𝑥_𝑛,𝑇^𝑛₁𝑦_𝑛,𝛼_𝑛,𝑝=𝑐. (19) It follows from Lemma 5 that lim_𝑛→∞𝑑𝑥_𝑛,𝑇^𝑛₁𝑦_𝑛=0. (20) On the other hand, since 𝑑𝑥_𝑛,𝑝≤𝑑𝑥_𝑛,𝑇^𝑛₁𝑦_𝑛+𝑑𝑇^𝑛₁𝑦_𝑛𝑥,𝑝≤𝑑_𝑛,𝑇^𝑛₁𝑦_𝑛+𝑑𝑦_𝑛,𝑝+𝜇_𝑛𝜙₁𝑑𝑦_𝑛,𝑝+𝜉_𝑛≤𝑑𝑥_𝑛,𝑇^𝑛₁𝑦_𝑛+1+𝜇_𝑛𝑀^∗𝑑𝑦_𝑛,𝑝+𝜉_𝑛, (21) we have liminf_𝑛→∞𝑑(𝑦_𝑛,𝑝)≥𝑐. Combined with (16), it yields that lim_𝑛→∞𝑑𝑦_𝑛,𝑝=𝑐. (22) This implies that lim_𝑛→∞𝑑𝑊𝑥_𝑛,𝑇^𝑛₂𝑥_𝑛,𝛽_𝑛,𝑝=𝑐. (23) Since 𝑑𝑇^𝑛₂𝑥_𝑛,𝑝≤𝑑𝑥_𝑛,𝑝+𝜇_𝑛𝜙₂𝑑𝑥_𝑛,𝑝+𝜉_𝑛≤1+𝜇_𝑛𝑀^∗𝑑𝑥_𝑛,𝑝+𝜉_𝑛, (24) we have limsup_𝑛→∞𝑑𝑇^𝑛₂𝑥_𝑛,𝑝≤𝑐. (25) So, it follows from (25) and Lemma 5 that lim_𝑛→∞𝑑𝑥_𝑛,𝑇^𝑛₂𝑥_𝑛=0. (26) Observe that 𝑑𝑥_𝑛,𝑇^𝑛₁𝑥_𝑛≤𝑑𝑥_𝑛,𝑇^𝑛₁𝑦_𝑛+𝑑𝑇^𝑛₁𝑦_𝑛,𝑇^𝑛₁𝑥_𝑛≤𝑑𝑥_𝑛,𝑇^𝑛₁𝑦_𝑛+𝑑𝑦_𝑛,𝑥_𝑛+𝜇_𝑛𝜙₁𝑑𝑦_𝑛,𝑥_𝑛+𝜉_𝑛=𝑑𝑥_𝑛,𝑇^𝑛₁𝑦_𝑛+1+𝜇_𝑛𝑀^∗𝑑𝑦_𝑛,𝑥_𝑛+𝜉_𝑛, (27) where 𝑑𝑦_𝑛,𝑥_𝑛=𝑑𝑊𝑥_𝑛,𝑇^𝑛₂𝑥_𝑛,𝛽_𝑛,𝑥_𝑛≤𝛽_𝑛𝑑𝑥_𝑛,𝑇^𝑛₂𝑥_𝑛. (28) It follows from (26) that lim_𝑛→∞𝑑𝑦_𝑛,𝑥_𝑛=0. (29) Thus, from (20), (27), and (29), we have lim_𝑛→∞𝑑𝑥_𝑛,𝑇^𝑛₁𝑥_𝑛=0. (30) In addition, since 𝑑𝑥_𝑛+1,𝑥_𝑛=𝑑𝑊𝑥_𝑛,𝑇^𝑛₁𝑦_𝑛,𝛼_𝑛,𝑥_𝑛≤𝛼_𝑛𝑑𝑥_𝑛,𝑇^𝑛₁𝑦_𝑛, (31) from (20), we have lim_𝑛→∞𝑑𝑥_𝑛+1,𝑥_𝑛=0. (32) Finally, since 𝑑𝑥_𝑛,𝑇₁𝑥_𝑛≤𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑇₁^𝑛+1𝑥_𝑛+1+𝑑𝑇₁^𝑛+1𝑥_𝑛+1,𝑇₁^𝑛+1𝑥_𝑛+𝑑𝑇₁^𝑛+1𝑥_𝑛,𝑇₁𝑥_𝑛≤(1+𝐿)𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑇₁^𝑛+1𝑥_𝑛+1+𝐿𝑑𝑇^𝑛₁𝑥_𝑛,𝑥_𝑛, (33) it follows from (30) and (32) that lim_𝑛→∞𝑑𝑥_𝑛,𝑇₁𝑥_𝑛=0. (34) Similarly, we also can show that lim_𝑛→∞𝑑𝑥_𝑛,𝑇₂𝑥_𝑛=0. (35)
Step 3. Now we prove that the sequence {𝑥_𝑛}Δ-converges to a common fixed point of 𝐹(𝑇₁)∩𝐹(𝑇₂).
In fact, since, for each 𝑝∈𝐹, lim_𝑛→∞𝑑(𝑥_𝑛,𝑝) exists, this implies that the sequence {𝑑(𝑥_𝑛,𝑝)} is bounded, so is the sequence {𝑥_𝑛}. Hence, by virtue of Lemma 3, {𝑥_𝑛} has a unique asymptotic center 𝐴_𝐾({𝑥_𝑛})={𝑥}.
Let {𝑢_𝑛} be any subsequence of {𝑥_𝑛} with 𝐴_𝐾({𝑢_𝑛})={𝑢}. It follows from (34) that lim_𝑛→∞𝑑𝑢_𝑛,𝑇₁𝑢_𝑛=0. (36) Now, we show that 𝑢∈𝐹(𝑇₁). For this, we define a sequence {𝑧_𝑛} in 𝐾 by 𝑧_𝑘=𝑇^𝑘₁𝑢. So, we calculate 𝑑𝑧_𝑘,𝑢_𝑛≤𝑑𝑇^𝑘₁𝑢,𝑇^𝑘₁𝑢_𝑛+𝑑𝑇^𝑘₁𝑢_𝑛,𝑇₁^𝑘−1𝑢_𝑛+⋯+𝑑𝑇₁𝑢_𝑛,𝑢_𝑛≤𝑑𝑢,𝑢_𝑛+𝜇_𝑛𝜙₁𝑑𝑢,𝑢_𝑛+𝜉_𝑛+^𝑘_𝑖=1𝑑𝑇^𝑖₁𝑢_𝑛,𝑇₁^𝑖−1𝑢_𝑛≤1+𝜇_𝑛𝑀^∗𝑑𝑢,𝑢_𝑛+𝜉_𝑛+^𝑘_𝑖=1𝑑𝑇^𝑖₁𝑢_𝑛,𝑇₁^𝑖−1𝑢_𝑛. (37) Since 𝑇₁ is uniformly 𝐿-Lipschitzian, from (37) we have 𝑑𝑧_𝑘,𝑢_𝑛≤1+𝜇_𝑛𝑀^∗𝑑𝑢,𝑢_𝑛+𝜉_𝑛+𝑘𝐿𝑑𝑇₁𝑢_𝑛,𝑢_𝑛. (38) Taking limsup on both sides of the previous estimate and using (36), we have 𝑟𝑧_𝑘,𝑢_𝑛=limsup_𝑛→∞𝑑𝑧_𝑘,𝑢_𝑛≤limsup_𝑛→∞𝑑𝑢,𝑢_𝑛=𝑟𝑢,𝑢_𝑛. (39) Since 𝐴_𝐾({𝑢_𝑛})={𝑢}, by the definition of asymptotic center 𝐴_𝐾({𝑢_𝑛}) of a bounded sequence {𝑢_𝑛} with respect to 𝐾⊂𝑋 and (8), this implies that 𝑟(𝑧_𝑘,{𝑢_𝑛})=𝑟(𝑢,{𝑢_𝑛}), for all 𝑘≥1. Therefore, |𝑟(𝑧_𝑘,{𝑢_𝑛})−𝑟(𝑢,{𝑢_𝑛})|→0 as 𝑘→∞. It follows from Lemma 6 that lim_𝑘→∞𝑇^𝑘₁𝑢=𝑢. As 𝑇₁ is uniformly continuous, 𝑇₁𝑢=𝑇₁(lim_𝑘→∞𝑇^𝑘₁𝑢)=lim_𝑘→∞𝑇₁^𝑘+1𝑢=𝑢. That is, 𝑢∈𝐹(𝑇₁). Similarly, we also can show that 𝑢∈𝐹(𝑇₂). Hence, 𝑢 is the common fixed point of 𝑇₁ and 𝑇₂. Reasoning as previously mentioned by utilizing the uniqueness of asymptotic centers, we get that 𝑥=𝑢. Since {𝑢_𝑛} is an arbitrary subsequence of {𝑥_𝑛}, 𝐴({𝑢_𝑛})={𝑢} for all subsequence {𝑢_𝑛} of {𝑥_𝑛}. This proves that {𝑥_𝑛}Δ-converges to a common fixed point of 𝑇₁ and 𝑇₂. This completes the proof.

The following theorem can be obtained from Theorem 7 immediately.

Theorem 8. Let 𝐾 be a nonempty closed convex subset of a complete uniformly convex hyperbolic space 𝑋 with monotone modulus of uniform convexity 𝜂. Let 𝑇_𝑖∶𝐾→𝐾,  𝑖=1,2, be asymptotically nonexpansive mappings with sequence {𝑡_𝑖𝑛}⊂[1,∞) satisfying lim_𝑛→∞𝑡_𝑖𝑛=1. Assume that 𝐹∶=⋂²_𝑖=1𝐹(𝑇_𝑖)≠∅; for arbitrarily chosen 𝑥₁∈𝐾, {𝑥_𝑛} is defined as follows: 𝑥_𝑛+1=𝑊𝑥_𝑛,𝑇^𝑛₁𝑦_𝑛,𝛼_𝑛,𝑦_𝑛=𝑊𝑥_𝑛,𝑇^𝑛₂𝑥_𝑛,𝛽_𝑛, (40) where {𝑡_𝑖𝑛}, 𝑖=1,2, {𝛼_𝑛} and {𝛽_𝑛} satisfy the following conditions: (1) ∑^∞_𝑛=1(𝑡_𝑖𝑛−1)<∞,𝑖=1,2;  (2)there exist constants 𝑎,𝑏∈(0,1) with 0<𝑏(1−𝑎)≤1/2 such that {𝛼_𝑛}⊂[𝑎,𝑏] and {𝛽_𝑛}⊂[𝑎,𝑏]. Then, the sequence {𝑥_𝑛} defined in (40) Δ-converges to a common fixed point of 𝐹∶=⋂²_𝑖=1𝐹(𝑇_𝑖).

Proof. Take 𝜙_𝑖(𝑡)=𝑡,𝑡≥0,𝜉_𝑖𝑛=0,𝜇_𝑖𝑛=𝑡_𝑖𝑛−1 in Theorem 7. Since all conditions in Theorem 7 are satisfied, it follows from Theorem 7 that the sequence {𝑥_𝑛}Δ-converges to a common fixed point of 𝐹∶=⋂²_𝑖=1𝐹(𝑇_𝑖).
This completes the proof of Theorem 8.

Acknowledgments

The authors would like to express their thanks to the editors and the referees for their helpful comments and suggestions. This work is supported by Scientific Research Fund of Sichuan Provincial Education Department (no. 11ZA222) and the Natural Science Foundation of Yibin University (no. 2012S07).