△ convergence for mixed type total asymptotically nonexpansive mappings in hyperbolic spaces

R E S E A R C H

Open Access

-convergence for mixed-type total

asymptotically nonexpansive mappings in

hyperbolic spaces

Li-Li Wan

*_{Correspondence:}

[email protected] School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China

Abstract

In this paper, we prove some-convergence theorems in a hyperbolic space. A mixed Agarwal-O’Regan-Sahu type iterative scheme for approximating a common fixed point of total asymptotically nonexpansive mappings is constructed. Our results extend some results in the literature.

MSC: 47H09; 49M05

Keywords: total asymptotically nonexpansive mappings; hyperbolic space; -convergence; mixed Agarwal-O’Regan-Sahu type iterative scheme

1 Introduction and preliminaries

In this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach []. Concretely, (X,d,W) is called a hyperbolic space if (X,d) is a metric space andW:X× X×[, ]→Xis a function satisfying

(I) ∀x,y,z∈X,∀λ∈[, ],d(z,W(x,y,λ))≤( –λ)d(z,x) +λd(z,y); (II) ∀x,y∈X,∀λ,λ∈[, ],d(W(x,y,λ),W(x,y,λ)) =|λ–λ| ·d(x,y);

(III) ∀x,y∈X,∀λ∈[, ],W(x,y,λ) =W(y,x, ( –λ));

(IV) ∀x,y,z,w∈X,∀λ∈[, ],d(W(x,z,λ),W(y,w,λ))≤( –λ)d(x,y) +λd(z,w).

If a space satisfies only (I), it coincides with the convex metric space introduced by Taka-hashi []. The concept of hyperbolic spaces in [] is more restrictive than the hyperbolic type introduced by Goebel [] since (I)-(III) together are equivalent to (X,d,W) being a space of hyperbolic type in []. But it is slightly more general than the hyperbolic space defined in Reich [] (see []). This class of metric spaces in [] covers all normed linear spaces,R-trees in the sense of Tits, the Hilbert ball with the hyperbolic metric (see []), Cartesian products of Hilbert balls, Hadamard manifolds (see [, ]) andCAT() spaces in the sense of Gromov (see []). A thorough discussion of hyperbolic spaces and a detailed treatment of examples can be found in [] (see also [–]).

A hyperbolic space isuniformly convex[] if foru,x,y∈X,r>  and∈(, ], there existsδ∈(, ] such that

d

W

x,y, 

,u

≤( –δ)r,

provided thatd(x,u)≤r,d(y,u)≤randd(x,y)≥r.

A mapη: (,∞)×(, ]→(, ] is calledmodulus of uniform convexityifδ=η(r,) for givenr> . Besides,ηismonotoneif it decreases withr(for a fixed), that is,

η(r,)≤η(r,), ∀r≥r> .

A subsetCof a hyperbolic spaceXisconvexifW(x,y,λ)∈Cfor allx,y∈Candλ∈[, ]. Let (X,d) be a metric space, and letCbe a nonempty subset ofX. Recall thatT:C→C

is said to be a ({νn},{μn},ζ)-total asymptotically nonexpansive mappingif there exist

non-negative sequences{νn},{μn}withνn→,μn→ and a strictly increasing continuous

functionζ: [,∞)→[,∞) withζ() =  such that

dTnx,Tny≤d(x,y) +νnζ

d(x,y)+μn, ∀n≥,x,y∈C. ()

It is well known that each nonexpansive mapping is an asymptotically nonexpansive map-ping and each asymptotically nonexpansive mapmap-ping is a ({νn},{μn},ζ)-total

asymptoti-cally nonexpansive mapping.

T:C→Cis said to beuniformly L-Lipschitzianif there exists a constantL>  such that

dTn_x,Tn_{y≤Ld(x,y), ∀n≥,x,y∈C.}

The following iteration process is a translation of the mixed Agarwal-O’Regan-Sahu type iterative scheme introduced in [] from Banach spaces to hyperbolic spaces. The iteration rate of convergence is similar to the Picard iteration process and faster than other fixed point iteration processes. Besides, it is independent of Mann and Ishikawa iteration pro-cesses.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈C,

xn+=W(Snxn,Tnyn,αn), n≥,

yn=W(S_nxn,T_nxn,βn),

()

whereCis a nonempty closed and convex subset of a complete uniformly convex hyper-bolic spaceXwith monotone modulus of uniform convexityη.Ti:C→C,i= , , is a uniformlyLi-Lipschitzian and ({νn(i)},{μ(ni)},ζ(i))-total asymptotically nonexpansive

map-ping, andSi:C→C, i= , , is a uniformlyLi-Lipschitzian and ({νn(i)},{μn(i)},ζ(i))-total

asymptotically nonexpansive mapping such that the following conditions are satisfied:

() ∞_n=ν(ni)<∞,

∞

n=μ (i)

n <∞,

∞

n=ν (i)

n <∞,

∞

n=μ (i)

n <∞,i= , ;

() There exists a constantM> such thatζ(i)_(r)≤Mr,ζ(i)_{(r)≤Mr,∀r≥,i= , .} Remark  Without loss of generality, we can assume thatTi:C→CandSi:C→C,i=

, , both are uniformlyL-Lipschitzian and ({νn},{μn},ζ)-total asymptotically

nonexpan-sive mappings satisfying conditions () and (). In fact, lettingνn=max{νn(i),νn(i),i= , }, μn=max{μ(ni),μ(ni),i= , },L=max{Li,Li,i= , } andζ =max{ζ(i),ζ(i),i= , }, thenSi

andTi,i= , , are the required mappings.

in aCAT() space using the mixed Agarwal-O’Regan-Sahu type iterative scheme. More precisely, one of the results is as follows.

Theorem  [] Let C be a bounded closed and convex subset of a complete CAT()

space X. Let Ti : C →C and Si :C → C, i= , , be uniformly L-Lipschitzian and

({νn},{μn},ζ)-total asymptotically nonexpansive mappings.IfF:= 

i=F(Ti)∩F(Si)=∅ and the following conditions are satisfied:

(i) ∞_n=νn<∞and

∞

n=μn<∞;

(ii) there exist constantsa,b∈(, )with <b( –a)≤_ such that{αn} ⊂[a,b];

(iii) there exists a constantM> such thatζ(r)≤Mr,r≥; (iv) d(x,Tiy)≤d(Six,Tiy)for allx,y∈Candi= , ,

then the sequence{xn}defined by()-converges to a common fixed point of Tiand Si,

i= , .

Theorem  can be viewed as a improvement and extension of several well-known results in Banach spaces andCAT() spaces, such as [] and []. Our purpose of this paper is to extend Theorem  from theCAT() spaces setting to the general setup of uniformly convex hyperbolic spaces.

Let{xn}be a bounded sequence in a hyperbolic spaceX. Forx∈X, we define

rx,{xn}=lim sup n→∞

d(x,xn).

Theasymptotic radius r({xn})of {xn}is given by

r{xn}

=infrx,{xn}

:x∈X.

Theasymptotic radius rC({xn})of {xn}with respect to C⊂Xis given by

rC{xn}=infrx,{xn}:x∈C.

Theasymptotic center A({xn})of {xn}is the set

A{xn}

=x∈X:rx,{xn}

=r{xn}

.

Theasymptotic center AC({xn})of {xn}with respect to C⊂Xis the set

AC{xn}=x∈C:rx,{xn}=rC{xn}.

Recall that a sequence{xn}inXis said to-converge to x∈Xifxis the unique asymptotic center of{un}for every subsequence{un}of{xn}. In this case, we callxthe-limit of {xn}.

The following lemmas are important in our paper.

Lemma [] Let(X,d,W)be a uniformly convex hyperbolic space with monotone modu-lus of uniform convexityη.Let x∈X and{αn}be a sequence in[a,b]for some a,b∈(, ).If

{xn}and{yn}are sequences in X such thatlim sup_n→∞d(xn,x)≤c,lim sup_n→∞d(yn,x)≤c andlimn→∞d(W(xn,yn,αn),x) =c for some c≥,then

lim

n→∞d(xn,yn) = .

Lemma [] Let C be a nonempty closed convex subset of a uniformly convex hyperbolic space,and let{xn}be a bounded sequence in C such that A({xn}) ={y}and r({xn}) =ρ.If

{ym}is another sequence in C such thatlimm→∞r(ym,{xn}) =ρ,thenlimm→∞ym=y.

Lemma [] Let{an},{bn}and{cn}be sequences of nonnegative numbers such that

an+≤( +bn)an+cn, ∀n≥.

If∞_n=bn<∞and_n∞_=cn<∞,thenlimn→∞anexists.

2 Main results

In this section, we prove our main theorems.

Theorem  Let C be a nonempty closed and convex subset of a complete uniformly con-vex hyperbolic space X with monotone modulus of uniform concon-vexity η.Let Ti:C→C and Si:C→C, i= , ,be uniformly L-Lipschitzian and({νn},{μn},ζ)-total asymptoti-cally nonexpansive mappings.IfF:=_i=F(Ti)∩F(Si)=∅and the following conditions are satisfied:

(i) ∞_n=νn<∞and

∞

n=μn<∞;

(ii) there exist constantsa,b∈(, )such that{αn} ⊂[a,b];

(iii) there exists a constantM> such thatζ(r)≤Mr,r≥; (iv) d(x,Tiy)≤d(Six,Tiy)for allx,y∈Candi= , ,

then the sequence{xn}defined by()-converges to a common fixed point of Tiand Si, i= , .

Proof We divide our proof into three steps.

Step . In the sequel, we shall show that for eachp∈F,

lim

n→∞d(xn,p) and nlim→∞d(xn,F) exist. ()

In fact, by conditions (), (), (I) and (iii), one gets

d(yn,p) =d

WS_nxn,Tnxn,βn

,p

≤( –βn)d

Sn_xn,p+βndT_nxn,p

≤( –βn)

d(xn,p) +νnζ

d(xn,p)+μn

+βn

d(xn,p) +νnζ

d(xn,p)+μn

=d(xn,p) +νnζ

d(xn,p)

+μn

and

d(xn+,p) =dWS_nxn,T_nyn,αn,p

≤( –αn)d

Sn_xn,p+αndT_nyn,p

≤( –αn)

d(xn,p) +νnζ

d(xn,p)+μn

+αn

d(yn,p) +νnζ

d(yn,p)+μn

≤( –αn)

( +νnM)d(xn,p) +μn

+αn

( +νnM)d(yn,p) +μn

. ()

Combining () and (), one has

d(xn+,p)≤( +σn)d(xn,p) +ξn, ∀n≥ andp∈F, ()

and

d(xn+,F)≤( +σn)d(xn,F) +ξn, ∀n≥, ()

whereσn=νnM(+αn(+νnM)),ξn= (+αn(+νnM))μn. Furthermore, using condition (i),

one has

∞

n=

σn<∞ and

∞

n=

ξn<∞. ()

Consequently, a combination of (), (), () and Lemma  shows that () is proved. Step . We claim that

lim

n→∞d(xn,Tixn) =  and nlim→∞d(xn,Sixn) = , i= , . ()

In fact, it follows from () thatlim_n→∞d(xn,p) exists for each givenp∈F. Without loss

of generality, we assume that

lim

n→∞d(xn,p) =c≥. ()

By () and (), one has

lim inf

n→∞ d(yn,p)≤lim sup_n→∞ d(yn,p)≤nlim→∞

( +νnM)d(xn,p) +μn

=c. ()

Noting

dT_nyn,p=dT_nyn,T_np

≤d(yn,p) +νnζ

d(yn,p)

+μn

≤( +νnM)d(yn,p) +μn, ∀n≥,

and

dSn_xn,p

=dSn_xn,Snp

≤d(xn,p) +νnζ

d(xn,p)+μn

by () and (), one has

lim sup n→∞

dT_nyn,p≤c and lim sup n→∞

dSn_xn,p≤c. ()

Besides, by () one gets

d(xn+,p) =d

WSn

xn,Tnyn,αn

,p≤( +σn)d(xn,p) +ξn,

which yields that

lim n→∞d

WSn

xn,Tnyn,αn

,p=c. ()

Now, by (), () and Lemma , we have

lim n→∞d

Sn

xn,Tnyn

= . ()

Using the same method, we can also have that

lim n→∞d

Sn_xn,T_nxn= . ()

It follows from (), () and condition (iv) that

lim n→∞d

xn,T_nyn≤ lim n→∞d

S_nxn,T_nyn=  ()

and

lim n→∞d

xn,T_nxn≤ lim n→∞d

Sn_xn,T_nxn= . ()

By virtue of (), one has

dyn,Sn_xn=dWSn_xn,T_nxn,βn

,Sn_xn

≤βndT_nxn,Sn_xn→ asn→ ∞. ()

Because we have

d(xn,yn)≤dxn,T_nxn+dT_nxn,Sn_xn+dSn_xn,yn,

it follows from (), () and () that

lim

n→∞d(xn,yn) = . ()

Combining () and (), one obtains

dxn,Tnxn

≤dxn,Tnyn

+dT_nyn,Tnxn

≤dxn,T_nyn+d(xn,yn) +νnζ

d(xn,yn)+μn

Moreover, it follows from () and () that

dSn_xn,T_nxn≤dS_nxn,T_nyn+dT_nyn,T_nxn

≤dS_nxn,T_nyn+Ld(yn,xn)→ asn→ ∞. ()

This jointly with () and () yields that

dSn

xn,xn

≤dSn

xn,Tnxn

+dTn

xn,xn

→ asn→ ∞,

and

d(xn+,xn)≤d

WSn_xn,Tnyn,αn

,xn

≤( –αn)dS_nxn,xn+αndT_nyn,xn→ asn→ ∞. ()

Now by (), () and (), for eachi= , , one gets

d(xn,Tixn)≤d(xn,xn+) +dxn+,T_in+xn+

+dT_in+xn+,T_in+xn

+dT_in+xn,Tixn

≤( +L)d(xn,xn+) +d

xn+,Tin+xn+

+LdT_inxn,xn→ asn→ ∞. ()

For eachi= , , the combination of (), (), (), () and (iv) yields that

d(xn,Sixn)≤d

xn,Tinxn

+dSixn,Tinxn

≤dxn,Tinxn

+dSn_ixn,Tinxn

→ asn→ ∞. ()

Therefore, () is proved.

Step . Now we are in a position to prove the -convergence of {xn}. Since{xn} is bounded, by Lemma , it has a unique asymptotic centerAC({xn}) ={x∗}. Let{un}be any subsequence of{xn}withAC({un}) ={u}, then by () and (), for eachi= , , we have

lim

n→∞d(un,Tiun) =  and d(un,Siun) = . ()

We claim thatu∈F. In fact, we define a sequence{zm}inCbyzm=T_mu. Then one has

d(zm,un)≤dT_mu,T_mun+dT_mun,T_m–un+· · ·+d(Tun,un)

≤d(u,un) +νnζ

d(u,un)

+μn+Ld(Tun,un) +· · ·+d(Tun,un).

By (), one gets

lim sup

n→∞ d(zm,un)≤lim supn→∞ d(u,un) =r

u,{un}

,

which yields that

Lemma  shows thatlimm→∞Tmu=u. BecauseTis uniformly continuous, we have

Tu=T

lim

m→∞T

m

 u

= lim

m→∞T

m+  u=u.

Hence,u∈F(T). Using the same method, we can prove thatu∈F. By the uniqueness of asymptotic centers, we get thatx∗=u. It implies thatx∗is the unique asymptotic center of{un}for each subsequence{un}of{xn}, that is,{xn} -converges tox∗∈F. The proof

is completed.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

Supported by General Project of Educational Department in Sichuan (No. 13ZB0182) and Doctor Research Foundation of Southwest University of Science and Technology (No. 11zx7130).

Received: 30 August 2013 Accepted: 22 October 2013 Published:22 Nov 2013

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