# Fixed Point Approximation of Generalized Nonexpansive Mappings in Hyperbolic Spaces

Authors: Jong Kyu Kim; Ramesh Prasad Pathak; Samir Dashputre; Shailesh Dhar Diwan; Rajlaxmi Gupta
Journal: International Journal of Mathematics and Mathematical Sciences
 (2015)

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## Abstract

We prove strong and Δ-convergence theorems for generalized nonexpansive mappings in uniformly convex hyperbolic spaces using S-iteration process due to Agarwal et al. As uniformly convex hyperbolic spaces contain Banach spaces as well as CAT(0) spaces, our results can be viewed as extension and generalization of several well-known results in Banach spaces as well as CAT(0) spaces.


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## Body

## 1. Introduction

Let(X,d) be a metric space and let C be a nonempty subset of X. A mapping T:C→C is said to be as follows:(i)
nonexpansive ifd(Tx,Ty)≤d(x,y) for all x,y∈C;(ii)
quasi-nonexpansive ifd(Tx,Tp)≤d(x,p) for all x∈C and p∈F(T), where F(T)={x∈C:Tx=x} denotes the set of fixed points of T.We know that there exist many generalizations of nonexpansive and quasi-nonexpansive mappings. Garcia-Falset et al. [1] introduced two generalizations of nonexpansive mappings which in turn include Suzuki generalized nonexpansive mappings (see [2]).Definition 1 (see [1]).
LetT be a mapping defined on a subset C of metric space X and μ≥1. Then T is said to satisfy the condition (Eμ), if, for all x,y∈C, (1)d(x,Ty)≤μd(x,Tx)+d(x,y).T is said to satisfy the condition (E) whenever T satisfies the condition (Eμ) for some μ≥1.Definition 2 (see [1]).
LetT be a mapping defined on a subset C of a metric space X and λ∈(0,1). Then T is said to satisfy the condition (Cλ) if, for all x,y∈C, (2)λd(x,Tx)≤d(x,y)⟹d(Tx,Ty)≤d(x,y).In the case0<λ1<λ2<1, then the condition (Cλ1) implies the condition (Cλ2). Suzuki (see [2]) said that T satisfies the condition (C), when λ=1/2.The following example shows that the class of mappings satisfying the conditions(E) and (Cλ), for some λ∈(0,1), is larger than the class of mappings satisfying the condition (C).Example 3 (see [1]).
For a givenλ∈(0,1), define a mapping T on [0,1] by (3)Tx=x2,ifx≠1,1+λ2+λ,ifx=1.Then the mapping T satisfies the condition (Cλ) but it fails the condition (Cλ1), whenever 0<λ1<λ. Moreover, T satisfies the condition (Eμ) for μ=(2+λ)/2.The basic properties and details of CAT(0) spaces can be found in the literature [3–5]. In [6], Lim introduced a concept of convergence in a general metric space which is called “Δ-convergence.” In 2008, Kirk and Panyanak [7] specialized Lim’s concept to CAT(0) spaces and showed that many Banach space results involving weak convergence have precise analogs in this setting. Since then, the existence problem and the Δ-convergence problem of iterative sequences to a fixed point for various classes of nonexpansive mappings in the frame work of CAT(0) spaces have been rapidly developed (see [1, 8–12]).In [13], Leustean proved that CAT(0) spaces are uniformly convex hyperbolic spaces with modulus of uniform convexity η(r,ε)=ε2/8 quadratic in ε. Thus, the class of uniformly convex hyperbolic spaces are a natural generalization of both uniformly convex Banach spaces and CAT(0) spaces.Throughout this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [14]. It is noted that they are different from Gromov hyperbolic spaces [15] or from other notions of hyperbolic spaces that can be found in literature (see [16–19]).A hyperbolic space(X,d,W) is a metric space (X,d) together with a convexity mapping W:X2×[0,1]→X satisfying(W1)d(u,W(x,y,α))≤αd(u,x)+(1-α)d(u,y);(W2)d(W(x,y,α),W(x,y,β))=|α-β|d(x,y);(W3)W(x,y,α)=W(y,x,1-α);(W4)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 metric space is said to be aconvex metric space in the sense of Takahashi [20], where a triple (X,d,W) satisfy only (W1)  (see [21–23]). We get the notion of the space of hyperbolic type in the sense of Goebel and Kirk [16], where a triple (X,d,W) satisfies (W1)–(W3). The (W4) was already considered by Itoh [24] under the name of “condition III” and it is used by Reich and Shafrir [18] and Kirk [17] to define their notions of hyperbolic spaces.The class of hyperbolic spaces include normed spaces and convex subsets thereof, the Hilbert space ball equipped with the hyperbolic metric [25], Hadrmard manifold, and the CAT(0) spaces in the sense of Gromov (see [15]).Ifx,y∈X and λ∈[0,1], then we use the notation (1-λ)x⊕λy  for  W(x,y,λ). The following holds even for the more general setting of convex metric space [20]: for all x,y∈X and λ∈[0,1],(4)dx,1-λx⊕λy=λdx,y,d(y,(1-λ)x⊕λy)=(1-λ)d(x,y).A hyperbolic space(X,d,W) is uniformly convex [13] if, for any r>0 and ε∈(0,2], there exists δ∈(0,1] such that, for all a,x,y∈X,(5)d12x⊕12y,a≤(1-δ)r,provided d(x,a)≤r, d(y,a)≤r, and d(x,y)≥εr.A mappingη:(0,∞)×(0,2]→(0,1], providing such a δ=η(r,ε) for given r>0 and ε∈(0,2], is called a modulus of uniform convexity. We called that η ismonotone if it decreases with r for fix ε.Recently, Agarwal et al. [26] introduced S-iteration process as follows (see [27]).LetC be a convex subset of a linear space X and let T be a mapping from C into itself. Then the iterative sequence {xn} generated from x1∈C and defined by(6)xn+1=(1-αn)Txn+αnTyn,yn=(1-βn)xn+βnTxn,n∈N,where {αn} and {βn} are sequences in (0,1) satisfying the certain condition. It is observed that rate of convergence of S-iteration process is similar to the Picard iteration process but faster than the Mann iteration process for contraction mapping (see [26, 27]).The purpose of this paper is to prove the strong andΔ-convergence theorems for generalized nonexpansive mappings in uniformly convex hyperbolic spaces by using S-iteration process. Our results can be viewed as extension and generalization of several well-known results in Banach spaces as well as CAT(0) spaces [10–12, 28, 29].


## 2. Preliminaries

LetC be a nonempty subset of metric space X and let {xn} be any bounded sequence in C. Consider a continuous functional ra(·,{xn}):X→R+ defined by (7)ra(x,{xn})=limsup⁡n→∞d(xn,x),x∈X.Then, the infimum of ra(·,{xn}) over C is said to be theasymptotic radius of {xn} with respect to C and is denoted by ra(C,{xn}).A pointz∈C is said to be anasymptotic center of the sequence {xn} with respect to C if (8)raz,xn=inf⁡rax,xn:x∈C;the set of all asymptotic centers of {xn} with respect to C is denoted by AC(C,{xn}). This set may be empty or a singleton or contain infinitely many points.If the asymptotic radius and the asymptotic center are taken with respect toX, then these are simply denoted by ra(X,{xn})=ra({xn}) and AC(X,{xn})=AC({xn}), respectively. We know that, for x∈X, ra(x,{xn})=0 if and only if limn→∞xn=x.It is known that every bounded sequence has a unique asymptotic center with respect to each closed convex subset in uniformly convex Banach spaces and even CAT(0) spaces.The following lemma is due to Leuştean [30] and ensures that this property also holds in a complete uniformly convex hyperbolic space.Lemma 4 (see [30]).
Let(X,d,W) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Then every bounded sequence {xn} in X has a unique asymptotic center with respect to any nonempty closed convex subset C of X.Recall that a sequence{xn} in X is said to be Δ-convergent to x∈X, if x is the unique asymptotic center of {un} for every subsequence {un} of {xn}. In this case, we write Δ-limnxn=x and call x the Δ-limit of {xn}.Lemma 5 (see [31]).
Let(X,d,W) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Let x∈X and {tn} be a sequence in [a,b] for some a,b∈(0,1). If {xn} and {yn} are sequences in X such that (9)limsup⁡n→∞d(xn,x)≤c,limsup⁡n→∞d(yn,x)≤c,limn→∞d(W(xn,yn,tn),x)=c,for some c≥0, then we have limn→∞d(xn,yn)=0.


## 3. Main Results

We begin with the definition ofFejér monotone sequences.Definition 6.
LetC be a nonempty subset of hyperbolic space X and let {xn} be a sequence in X. Then {xn} is said to beFejér monotone with respect to C if for all x∈C and n∈N(10)d(xn+1,x)≤d(xn,x).Example 7.
LetC be a nonempty subset of hyperbolic space X and let T:C→C be a quasi-nonexpansive (in particular, nonexpansive) mapping such that F(T)≠∅. Then the sequence {xn} of Picard iteration isFejér monotone with respect to F(T).We can easily prove the following proposition.Proposition 8.
Let{xn} be a sequence in X and let C be a nonempty subset of X. Suppose that {xn} isFejér monotone with respect to C. Then we have the following: (1)
{xn} is bounded;(2)
the sequence{d(xn,p)} is decreasing and convergent for all p∈F(T).We now define S-iteration process in hyperbolic spaces:LetC be a nonempty closed convex subset of a hyperbolic space X and let T be a mapping of C into itself. For any x1∈C the sequence {xn} of S-iteration process is defined as(11)xn+1=W(Txn,Tyn,αn)yn=W(xn,Txn,βn),n∈N,where {αn} and {βn} are real sequences with 0<a≤αn, βn≤b<1.Lemma 9.
LetC be a nonempty closed convex subset of a hyperbolic space X and let T:C→C be a mapping which satisfies the condition (Cλ) for some λ∈(0,1). If {xn} is a sequence defined by (11), then {xn} isFejér monotone with respect to F(T).Proof.
Letp∈F(T). Then we have(12)λdp,Tp=0≤dp,yn,λdp,Tp=0≤dp,xn,for all n∈N. Since T satisfies the condition (Cλ), for some λ∈(0,1), we have (13)dTp,Tyn≤dp,yn,d(Tp,Txn)≤d(p,xn).Using (11), we have(14)dyn,p=d(W(xn,Txn,βn),p)≤(1-βn)d(xn,p)+βnd(Txn,p)≤d(xn,p).Again, using (11) and (14), we have (15)dxn+1,p=d(W(Txn,Tyn,αn),p)≤(1-αn)d(Txn,p)+αnd(Tyn,p)≤(1-αn)d(xn,p)+αnd(yn,p)=d(xn,p);that is, d(xn+1,p)≤d(xn,p) for all p∈F(T). Thus, {xn} isFejér monotone with respect to F(T).Lemma 10.
LetC be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity η and let T:C→C be a mapping which satisfies the conditions (Cλ) and (E) on C. If {xn} is a sequence defined by (11), then F(T) is nonempty if and only if {xn} is bounded and limn→∞d(xn,Txn)=0.Proof.
Suppose that the fixed point setF(T) is nonempty and p∈F(T). From Lemma 9, we know that {xn} isFejér monotone with respect to F(T). Hence, by Proposition 8, {xn} is bounded and limn→∞d(xn,p) exists. Let limn→∞d(xn,p)=c. Since (16)λdp,Tp=0≤dp,yn,λd(p,Tp)=0≤d(p,xn),for all n∈N, from the condition (Cλ), we have (17)dTp,Tyn≤dp,yn,d(Tp,Txn)≤d(p,xn).Therefore, (18)dTxn,p≤dTxn,Tp≤dxn,p,for all n∈N. Taking the limit supremum on both sides, we get (19)limsup⁡n→∞d(Txn,p)≤c.Similarly, we have (20)limsup⁡n→∞d(Tyn,p)≤c.Taking the limit supremum on both sides of (14), we have(21)limsup⁡n→∞d(yn,p)≤c.Since (22)c=limsup⁡n→∞d(xn+1,p)=limn→∞d(W(Txn,Tyn,αn),p),by using (19), (20), and Lemma 5, we get (23)limn→∞d(Txn,Tyn)=0.Next, we know that(24)dxn+1,Txn=d(W(Txn,Tyn,αn),Txn)≤bd(Tyn,Txn)⟶0,asn⟶∞.Hence, from (23) and (24), we have(25)dxn+1,Tyn≤d(xn+1,Txn)+d(Txn,Tyn)⟶0,asn⟶∞.Now we observe that(26)dxn+1,p≤d(xn+1,Tyn)+d(Tyn,p)≤d(xn+1,Tyn)+d(yn,p),which yields that (27)c≤liminf⁡n→∞d(yn,p).From the estimates of (21) and (27), we have that (28)limn→∞d(yn,p)=c.Thus, from (11), we have (29)limn→∞d(W(xn,Txn,βn),p)=c,which gives (30)limn→∞d(xn,Txn)=0.
Conversely, suppose that{xn} is bounded and limn→∞d(xn,Txn)=0. Let AC(C,{xn})={x}. Then, by Lemma 4, x∈C. As T satisfies the condition (Eμ) on C, there exists μ>1 such that (31)d(xn,Tx)≤μd(xn,Txn)+d(xn,x),which implies that (32)limsup⁡n→∞dxn,Tx≤limsup⁡n→∞{μd(xn,Txn)+d(xn,x)}=limsup⁡n→∞d(xn,x).By using the uniqueness of asymptotic center, Tx=x, so x is a fixed point ofT.Theorem 11.
LetC be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η and let T:C→C be a mapping which satisfies conditions (Cλ) and (E), for some λ∈(0,1) on C with F(T)≠∅. Then the sequence {xn} which is defined by (11), is Δ-convergent to a fixed point of T.Proof.
From Lemma10, we know that {xn} is a bounded sequence; therefore, {xn} has a Δ-convergent subsequence. We now prove that every Δ-convergent subsequence of {xn} has unique Δ-limt in F(T). For this, let u and v be Δ-limits of the subsequences {un} and {vn} of {xn}, respectively. By Lemma 4, AC(C,{un})={u} and AC(C,{vn})={v}. Since {un} is bounded sequence, by Lemma 10, limn→∞d(un,Tun)=0. We claim that u is a fixed point of T. Since T satisfies the condition (E), there exists a μ≥1 such that (33)d(un,Tu)≤μd(un,Tun)+d(un,u).Taking the limit supremum on both sides, we have(34)raun,Tu=limsup⁡n→∞d(un,Tu)≤limsup⁡n→∞{μd(un,Tun)+d(un,u)}≤limsup⁡n→∞d(un,u)=ra({un},u).Hence, we obtain (35)ra({un},Tu)≤ra({un},u).By uniqueness of the asymptotic center, Tu=u.
Similarly, we can prove thatTv=v. Thus, u and v are fixed points of  T. Now we show that u=v. If not, then by the uniqueness of asymptotic center,(36)limsup⁡n→∞dxn,u=limsup⁡n→∞d(un,u)<limsup⁡n→∞d(un,v)=limsup⁡n→∞d(xn,v)=limsup⁡n→∞d(vn,v)<limsup⁡n→∞dvn,u=limsup⁡n→∞d(xn,u),which is a contradiction. Hence u=v.Theorem 12.
LetC be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η and let T:C→C be a mapping which satisfies conditions (Cλ) and (E), for some λ∈(0,1) on C with F(T)≠∅. Then the sequence {xn} which is defined by (11) converges strongly to a fixed point ofT if and only if (37)liminf⁡n→∞D(xn,F(T))=0,where D(xn,F(T))=inf⁡x∈F(T)d(xn,x).Proof.
Necessity is obvious; we have to prove only sufficient part. First, we show that the fixed point setF(T) is closed; let {xn} be a sequence in F(T) which converges to some point z∈C. Since (38)λd(xn,Txn)=0≤d(xn,z),from the condition (Cλ), we have (39)d(xn,Tz)=d(Txn,Tz)≤d(xn,z).By taking the limit on both sides, we obtain (40)limn→∞d(xn,Tz)≤limn→∞d(xn,z)=0.In view of the uniqueness of the limit, we have z=Tz, so that F(T) is closed. Suppose that (41)liminf⁡n→∞D(xn,F(T))=0.Then, from (15) (42)Dxn+1,FT≤Dxn,FT;it follows from Lemma 9 and Proposition 8 that limn→∞D(xn,F(T)) exists. Hence we know that limn→∞D(xn,F(T))=0.
Consider a subsequence{xnk} of {xn} such that (43)d(xnk,pk)<12k,∀k≥1,where {pk} is in F(T). By Lemma 9, we have (44)d(xnk+1,pk)≤d(xnk,pk)<12k,which implies that(45)dpk+1,pk≤d(pk+1,xnk+1)+d(xnk+1,pk)<12k+1+12k<12k-1.This shows that {pk} is a Cauchy sequence. Since F(T) is closed, {pk} is a convergent sequence. Let limk→∞pk=p. Then, we know that {xn} converges to p. In fact, since (46)d(xnk,p)≤d(xnk,pk)+d(pk,p)⟶0ask⟶∞,we have (47)limk→∞d(xnk,p)=0.Since limn→∞d(xn,p) exists, the sequence {xn} is convergent to p.We recall the definition of condition(I) due to Senter and Doston [32].LetC be a nonempty subset of a metric space X. A mapping T:C→C is said to satisfy condition (I), if there is a nondecreasing function f[0,∞)→[0,∞) with f(0)=0, f(t)>0 for all t∈(0,∞) such that (48)d(x,Tx)≥f(D(x,F(T))),for all x∈C, where D(x,F(T))=inf⁡{d(x,p):p∈F(T)}.Theorem 13.
LetC be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η and let T:C→C be a mapping which satisfies conditions (Cλ) and (E), for some λ∈(0,1) on C. Moreover, T satisfies condition (I) with F(T)≠∅. Then the sequence {xn} which is defined by (11) converges strongly to some fixed point ofT.Proof.
As in the proof of Theorem12, it can be shown that F(T) is closed. Observe that, by Lemma 9, we have limn→∞d(xn,Txn)=0. It follows from condition (I) that (49)limn→∞f(D(xn,F(T)))≤limn→∞d(xn,Txn)=0.Therefore, we have (50)limn→∞f(D(xn,F(T)))=0.Since f:[0,∞]→[0,∞) is a nondecreasing mapping satisfying f(0)=0, f(t)>0 for all t∈(0,∞), we have (51)limn→∞D(xn,F(T))=0.Rest of the proof follows in lines of Theorem 12.Remark 14.
Our Theorems11, 12, and 13 improve and extend the previous well-known results from Banach spaces and CAT(0) spaces to more general class of uniformly convex hyperbolic spaces (see [10, 28, 29], in particular, Theorems 3.4 and 3.6 of [12]). In our results, we considered the faster iteration process to approximate the fixed point of underlying mapping T in the framework of uniformly convex hyperbolic spaces.



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