# Convergence Theorems for Total Asymptotically Nonexpansive Mappings in Hyperbolic Spaces

Authors: Liang-cai Zhao; Shih-sen Chang; Xiong Rui Wang
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

The purpose of this paper is to introduce the concept oftotal 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.


---

## Body

## 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
(1)d(W(x,y,12),u)≤(1-δ)r,
provided d(x,u)≤r, d(y,u)≤r and d(x,y)≥ϵr.A mapη:(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 mappingT:K→K is said to be nonexpansive, if
(2)d(Tx,Ty)≤d(x,y),∀x,y∈K.A mappingT:K→K is said to be asymptotically nonexpansive if there exists a sequence {kn}⊂[1,∞) with kn→1 such that
(3)d(Tnx,Tny)≤knd(x,y),∀n≥1,x,y∈K.A mappingT:K→K is said to be uniformly L-Lipschitzian if there exists a constant L>0 such that
(4)d(Tnx,Tny)≤Ld(x,y),∀n≥1,x,y∈K.Definition 1.
A mappingT:K→K is said to be ({μn},{ξn},ϕ)-total asymptotically nonexpansive, if there exist nonnegative sequences {μn},{ξn} with μn→0,  ξn→0 and a strictly increasing continuous function ϕ:[0,∞)→[0,∞) with ϕ(0)=0 such that
(5)d(Tnx,Tny)≤d(x,y)+μnϕ(d(x,y))+ξn,222222222222∀n≥1,x,y∈K.Remark 2.
From the definitions, it is clear that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence{kn=1} and each asymptotically nonexpansive mapping is a ({μn},{ξn},ϕ)-total asymptotically nonexpansive mapping with ξn=0, μn=kn-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 someΔ-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 conceptΔ-convergence in the general setup of hyperbolic spaces, we first collect some basic concepts.Let{xn} be a bounded sequence in a hyperbolic space X. For x∈X, we define a continuous functional r(·,{xn}):X→[0,∞) by
(6)r(x,{xn})=limsupn→∞d(x,xn).Theasymptotic radius  r({xn}) of {xn} is given by
(7)r({xn})=inf{r(x,{xn}):x∈X}.Theasymptotic centerAk({xn}) of a bounded sequence {xn} with respect to K⊂X is the set
(8)AK({xn})={x∈X:r(x,{xn})≤r(y,{xn}),∀y∈K}.This is the set of minimizers of the functionalr(·,{xn}). If the asymptotic center is taken with respect to X, then it is simply denoted by A({xn}). 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 {xn} in X has a unique asymptotic center with respect to any nonempty closed convex subset K of X.
Recall that a sequence{xn} in X is said to Δ-converge to x∈X if x is the unique asymptotic center of {un} for every subsequence {un} of {xn}. In this case, we write Δ-limn→∞xn=x and call x the Δ-limit of {xn}.
A mappingT:K→K is semicompact if every bounded sequence {xn}⊂K, satisfying d(xn,Txn)→0, has a convergent subsequence.Lemma 4 (see [8]).
Let{an}, {bn}, and {δn} be sequences of nonnegative real numbers satisfying
(9)an+1≤(1+δn)an+bn,∀n≥1.
If ∑n=1∞‍δn<∞ and ∑n=1∞‍bn<∞, then the limit limn→∞an exists. If there exists a subsequence {ani}⊂{an} such that ani→0, then limn→∞an=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 {αn} be a sequence in [a,b] for some a,b∈(0,1). If {xn} and {yn} are sequences in X such that
(10)limsupn→∞d(xn,x)≤c,limsupn→∞d(yn,x)≤c,limn→∞d(W(xn,yn,αn),x)=c,
for some c≥0, then limn→∞d(xn,yn)=0.Lemma 6 (see [11]).
LetK be a nonempty closed convex subset of uniformly convex hyperbolic space and {xn} a bounded sequence in K such that A({xn})={y} and r({xn})=ρ. If {ym} is another sequence in K such that limm→∞r(ym,{xn})=ρ, then limm→∞ym=y.


## 2. Main Results

Theorem 7.
LetK be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let Ti:K→K,i=1,2, be uniformly Li-Lipschitzian and ({μin},{ξin},ϕi)-total asymptotically nonexpansive mappings with sequences {μin} and {ξin} satisfying limn→∞μin=0, limn→∞ξin=0, and strictly increasing function ϕi:[0,∞)→[0,∞) with ϕi(0)=0,  i=1,2. Assume that F:=⋂i=12F(Ti)≠∅, and for arbitrarily chosen x1∈K, {xn} is defined as follows:
(11)xn+1=W(xn,T1nyn,αn),yn=W(xn,T2nxn,βn),
where {μin}, {ξin}, ϕi, i=1,2, {αn}, and {βn} satisfy the following conditions: (1)
∑n=1∞‍μin<∞,∑n=1∞‍ξin<∞,i=1,2;(2)
there exist constantsa,b∈(0,1) with 0<b(1-a)≤1/2 such that {αn}⊂[a,b] and {βn}⊂[a,b];(3)
there exist a constantM*>0 such that ϕi(r)≤M*r,r>0,i=1,2. Then, the sequence {xn} defined by (11) Δ-converges to a common fixed point of F:=⋂i=12F(Ti).Proof.
The proof of Theorem7 is divided into four steps.
Step
1. First, we prove that limn→∞d(xn,p) exists for each p∈F.
Setμn=max{μ1n,μ2n}, ξn=max{ξ1n,ξ2n}, and L=max{L1,L2}. Since ∑n=1∞‍μin<∞,∑n=1∞‍ξin<∞,  i=1,2, ∑n=1∞‍μn<∞,  ∑n=1∞‍ξn<∞. For any p∈F(T1)∩F(T2), by (11) we have
(12)d(xn+1,p)=d(W(xn,T1nyn,αn),p)≤(1-αn)d(xn,p)+αnd(T1nyn,p)≤(1-αn)d(xn,p)+αn[d(yn,p)+μnϕ1(d(yn,p))+ξn]≤(1-αn)d(xn,p)+αn[(1+μnM*)d(yn,p)+ξn],
where
(13)d(yn,p)=d(W(xn,T2nxn,βn),p)≤(1-βn)d(xn,p)+βnd(T2nxn,p)≤(1-βn)d(xn,p)+βn[d(xn,p)+μnϕ2(d(xn,p))+ξn]≤(1-βn)d(xn,p)+βn[(1+μnM*)d(xn,p)+ξn]≤(1+βnμnM*)d(xn,p)+βnξn.
Substituting (13) into (12), we have
(14)d(xn+1,p)≤(1-αn)d(xn,p)+αn[(1+μnM*)((1+βnμnM*)d(xn,p)+βnξn)+ξn]≤[1+(1+βn+βnμnM*)αnμnM*]d(xn,p)+(1+βn+βnμnM*)αnξn.
Applying Lemma 4 to the inequality, we get that limn→∞d(xn,p) exist for p∈F(T1)∩F(T2).
Step
2.We show that limn→∞d(xn,T1xn)=limn→∞d(xn,T2xn)=0.
For eachp∈F(T1)∩F(T2), from the proof of Step 1, we know that limn→∞d(xn,p) exists. We may assume that limn→∞d(xn,p)=c≥0. The case c=0 is trivial. Next, we deal with the case c>0. From (13), we have
(15)d(yn,p)≤(1+βnμnM*)d(xn,p)+βnξn.
Taking limsup on both sides in (15), we have
(16)limsupn→∞d(yn,p)≤c.
In addition, since
(17)d(T1nyn,p)≤d(yn,p)+μnϕ1(d(yn,p))+ξn≤(1+μnM*)d(yn,p)+ξn,
we have
(18)limsupn→∞d(T1nyn,p)≤c.
Since limn→∞d(xn+1,p)=c, it is easy to prove that
(19)limn→∞d(W(xn,T1nyn,αn),p)=c.
It follows from Lemma 5 that
(20)limn→∞d(xn,T1nyn)=0.
On the other hand, since
(21)d(xn,p)≤d(xn,T1nyn)+d(T1nyn,p)≤d(xn,T1nyn)+d(yn,p)+μnϕ1(d(yn,p))+ξn≤d(xn,T1nyn)+(1+μnM*)d(yn,p)+ξn,
we have liminfn→∞d(yn,p)≥c. Combined with (16), it yields that
(22)limn→∞d(yn,p)=c.
This implies that
(23)limn→∞d(W(xn,T2nxn,βn),p)=c.
Since
(24)d(T2nxn,p)≤d(xn,p)+μnϕ2(d(xn,p))+ξn≤(1+μnM*)d(xn,p)+ξn,
we have
(25)limsupn→∞d(T2nxn,p)≤c.
So, it follows from (25) and Lemma 5 that
(26)limn→∞d(xn,T2nxn)=0.
Observe that
(27)d(xn,T1nxn)≤d(xn,T1nyn)+d(T1nyn,T1nxn)≤d(xn,T1nyn)+d(yn,xn)+μnϕ1(d(yn,xn))+ξn=d(xn,T1nyn)+(1+μnM*)d(yn,xn)+ξn,
where
(28)d(yn,xn)=d(W(xn,T2nxn,βn),xn)≤βnd(xn,T2nxn).
It follows from (26) that
(29)limn→∞d(yn,xn)=0.
Thus, from (20), (27), and (29), we have
(30)limn→∞d(xn,T1nxn)=0.
In addition, since
(31)d(xn+1,xn)=d(W(xn,T1nyn,αn),xn)≤αnd(xn,T1nyn),
from (20), we have
(32)limn→∞d(xn+1,xn)=0.
Finally, since
(33)d(xn,T1xn)≤d(xn,xn+1)+d(xn+1,T1n+1xn+1)+d(T1n+1xn+1,T1n+1xn)+d(T1n+1xn,T1xn)≤(1+L)d(xn,xn+1)+d(xn+1,T1n+1xn+1)+Ld(T1nxn,xn),
it follows from (30) and (32) that
(34)limn→∞d(xn,T1xn)=0.
Similarly, we also can show that
(35)limn→∞d(xn,T2xn)=0.
Step
3. Now we prove that the sequence {xn}Δ-converges to a common fixed point of F(T1)∩F(T2).
In fact, since, for eachp∈F, limn→∞d(xn,p) exists, this implies that the sequence {d(xn,p)} is bounded, so is the sequence {xn}. Hence, by virtue of Lemma 3, {xn} has a unique asymptotic center AK({xn})={x}.
Let{un} be any subsequence of {xn} with AK({un})={u}. It follows from (34) that
(36)limn→∞d(un,T1un)=0.
Now, we show that u∈F(T1). For this, we define a sequence {zn} in K by zk=T1ku. So, we calculate
(37)d(zk,un)≤d(T1ku,T1kun)+d(T1kun,T1k-1un)+⋯+d(T1un,un)≤d(u,un)+μnϕ1(d(u,un))+ξn+∑i=1k‍d(T1iun,T1i-1un)≤(1+μnM*)d(u,un)+ξn+∑i=1k‍d(T1iun,T1i-1un).
Since T1 is uniformly L-Lipschitzian, from (37) we have
(38)d(zk,un)≤(1+μnM*)d(u,un)+ξn+kLd(T1un,un).
Taking limsup on both sides of the previous estimate and using (36), we have
(39)r(zk,{un})=limsupn→∞d(zk,un)≤limsupn→∞d(u,un)=r(u,{un}).
Since AK({un})={u}, by the definition of asymptotic center AK({un}) of a bounded sequence {un} with respect to K⊂X and (8), this implies that r(zk,{un})=r(u,{un}), for all k≥1. Therefore, |r(zk,{un})-r(u,{un})|→0 as k→∞. It follows from Lemma 6 that limk→∞T1ku=u. As T1 is uniformly continuous, T1u=T1(limk→∞T1ku)=limk→∞T1k+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 {un} is an arbitrary subsequence of {xn}, A({un})={u} for all subsequence {un} of {xn}. This proves that {xn}Δ-converges to a common fixed point of T1 and T2. This completes the proof.The following theorem can be obtained from Theorem7 immediately.Theorem 8.
LetK be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let Ti:K→K,  i=1,2, be asymptotically nonexpansive mappings with sequence {tin}⊂[1,∞) satisfying limn→∞tin=1. Assume that F:=⋂i=12F(Ti)≠∅; for arbitrarily chosen x1∈K, {xn} is defined as follows:
(40)xn+1=W(xn,T1nyn,αn),yn=W(xn,T2nxn,βn),
where {tin}, i=1,2, {αn} and {βn} satisfy the following conditions:(1)∑n=1∞‍(tin-1)<∞,i=1,2;(2)there exist constants a,b∈(0,1) with 0<b(1-a)≤1/2 such that {αn}⊂[a,b] and {βn}⊂[a,b]. Then, the sequence {xn} defined in (40) Δ-converges to a common fixed point of F:=⋂i=12F(Ti).Proof.
Takeϕi(t)=t,t≥0,ξin=0,μin=tin-1 in Theorem 7. Since all conditions in Theorem 7 are satisfied, it follows from Theorem 7 that the sequence {xn}Δ-converges to a common fixed point of F:=⋂i=12F(Ti).
This completes the proof of Theorem8.



---

*Source: 689765-2013-11-03.xml*