Another type of Mann iterative scheme for two mappings in a complete geodesic space

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R E S E A R C H

Open Access

Another type of Mann iterative scheme for

two mappings in a complete geodesic space

Yasunori Kimura and Koichi Nakagawa

*

*_{Correspondence:}

[email protected] Department of Information Science, Toho University, Miyama, Funabashi, Chiba 274-8510, Japan

Abstract

In this paper, we show a

-convergence theorem for a Mann iteration procedure in a

complete geodesic space with two quasinonexpansive and

-demiclosed

mappings. The proposed method is diﬀerent from known procedures with respect to the order of taking the convex combination.

1 Introduction

The fixed point approximation has been studied in a variety of ways and its results are useful for the other studies. In , Mann [] introduced an iteration procedure for ap-proximating fixed points of a nonexpansive mappingT in a Hilbert space. Later, Reich [] discussed this iteration procedure in a uniformly convex Banach space whose norm is Fréchet differentiable. In , Takahashi and Tamura [] considered an iteration proce-dure with two nonexpansive mappings and obtained weak convergence theorems for this procedure in a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. On the other hand, in , Dhompongsa and Panyanak [] proved the following theorem.

Theorem . Let C be a bounded closed convex subset of a completeCAT()space and T:C→C a nonexpansive mapping.For any initial point xin C,deﬁne the Mann iterative sequence{xn}by

xn+= ( –tn)xn⊕tnTxn, n= , , , . . . ,

where {tn} is a sequence in [, ], with the restrictions that

∞

n=tn diverges and

lim sup_n_→∞tn< .Then{xn}-converges to a ﬁxed point of T.

Further, in aCAT() space, Kimuraet al.[] proved the-convergence theorem for a family of nonexpansive mappings including the following scheme:

xn+= ( –αn)xn⊕αn

( –βn)Sxn⊕βnTxn

.

In a Hilbert spaceH, the following equality holds for anyx,y,z∈H:

αx+ ( –α)βy+ ( –β)z=γδx+ ( –δ)y+ ( –γ)z,

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whereα,β,γ,δ∈], [ such thatα=γ δandβ=γ( –δ)/( –γ δ). However, inCAT(κ) spaces withκ> , it does not hold in general, that is, the value of the convex combination taken twice depends on their order. Thus, the following formulas are diﬀerent in gen-eral:

xn+= ( –αn)xn⊕αn

( –βn)Sxn⊕βnTxn

,

xn+= ( –αn)

βnxn⊕( –βn)Sxn

⊕αn

( –βn)xn⊕βnTxn

.

()

In this paper, we show an analogous result to Theorem . using the iterative scheme () in a completeCAT() space with two quasinonexpansive and-demiclosed mappings. We also deal with the image recovery problem for two closed convex sets.

2 Preliminaries

LetXbe a metric space. Forx,y∈X, a mappingc: [,l]→Xis said to be a geodesic ifc

satisﬁesc() =x,c(l) =yandd(c(s),c(t)) =|s–t|for alls,t∈[,l]. An image [x,y] ofcis called a geodesic segment joiningxandy. Forr> ,Xis said to be anr-geodesic metric space if, for anyx,y∈Xwithd(x,y) <r, there exists a geodesic segment [x,y]. In particular, if a segment [x,y] is unique for anyx,y∈Xwithd(x,y) <r, thenXis said to be a uniquely

r-geodesic metric space. In what follows, we always assumed(x,y) <π/ for anyx,y∈X. Thus, we sayXis a geodesic metric space instead of aπ/-geodesic metric space. For the more general case, see [].

LetXbe a uniquely geodesic metric space. A geodesic triangle is deﬁned by(x,y,z) = [x,y]∪[y,z]∪[z,x]. LetMbe the two-dimensional unit sphere in R_{. For} _x_¯_,_y_¯_,_z_¯_∈_M_,

a triangle (x¯,y¯,z¯)⊂Mis called a comparison triangle of(x,y,z) ifd(x,y) =dM(x¯,y¯),

d(y,z) =dM(y¯,¯z),d(z,x) =dM(z¯,x¯). Further, for anyx,y∈Xandt∈], [, ifz∈[x,y] satis-ﬁesd(x,z) = ( –t)d(x,y) andd(z,y) =td(x,y), thenzis denoted byz=tx⊕( –t)y. A point

¯

z∈[x¯,y¯] is called a comparison point ofz∈[x,y] ifd(x,z) =dM(x¯,z¯).Xis said to be aCAT() space if, for anyp,q∈ (x,y,z)⊂Xand its comparison pointsp¯,q¯∈ (x¯,y¯,z¯)⊂M, the in-equalityd(p,q)≤dM(p¯,q¯) holds.

Let Xbe a geodesic metric space and{xn} a bounded sequence ofX. Forx∈X, we putr(x,{xn}) =lim supn→∞d(x,xn). The asymptotic radius of{xn}is deﬁned byr({xn}) =

infx∈Xr(x,{xn}). Further, the asymptotic center of{xn} is deﬁned byAC({xn}) ={x∈X:

r(x,{xn}) =r({xn})}. If, for any subsequences {xnk} of {xn}, AC({xnk}) ={x}, i.e., their

asymptotic center consists of the unique elementx, then we say{xn}-converges tox

and we denote it byxn–

x.

Let X be a metric space. A mappingT :X→X is said to be a nonexpansive if T

satisﬁes d(Tx,Ty)≤d(x,y) for any x,y∈X. The set of ﬁxed points ofT is denoted by

F(T) ={z∈X:Tz=z}. Further, a mappingT:X→XwithF(T)=∅is said to be a quasi-nonexpansive ifT satisﬁesd(Tx,z)≤d(x,z) for anyx∈Xandz∈F(T). Moreover,T is said to be-demiclosed if, for any bounded sequence {xn} ⊂X andx∈X satisfying d(xn,Txn)→ andxn–

x, we havex∈F(T).

3 Tools for the main results

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Theorem .(Kimura and Satô []) Let(x,y,z)be a geodesic triangle in aCAT()space such that d(x,y) +d(y,z) +d(z,x) < π.Let u=tx⊕( –t)y for some t∈[, ].Then

cosd(u,z)sind(x,y)≥cosd(x,z)sintd(x,y) +cosd(y,z)sin( –t)d(x,y).

Corollary .(Kimura and Satô []) Let(x,y,z)be a geodesic triangle in aCAT()space such that d(x,y) +d(y,z) +d(z,x) < π.Let u=tx⊕( –t)y for some t∈[, ].Then

cosd(u,z)≥tcosd(x,z) + ( –t)cosd(y,z).

Theorem .(Espínola and Fernández-León []) Let X be a completeCAT()space and

{xn}a sequence in X.If r({xn}) <π/,then the following hold. (i) AC({xn})consists of exactly one point;

(ii) {xn}has a-convergent subsequence.

Theorem .(Kimura and Satô []) Let X be a metric space and T a mapping from X into

itself.If T is a nonexpansive with F(T)=∅,then T is quasinonexpansive and-demiclosed.

The following lemmas are important properties of real numbers and they are easy to show. Thus, we omit the proofs.

Lemma . Letδ be a real number such that – <δ< and{bn}, {cn} real sequences

satisfyingδ≤bn≤,δ≤cn≤andlim infn→∞bncn≥.Thenlimn→∞bn=limn→∞cn= .

Lemma . Let s∈],∞[and{bn},{cn}bounded real sequences satisfying bn≤,s<cn

andlimn→∞bn/cn= .Thenlimn→∞bn= .

Lemma . Let{bn}and{cn}be bounded real sequences satisfyinglimn→∞(bn–cn) = .

Thenlim infn→∞bn=lim infn→∞cn.

4 The main result

In this section, we show the main result.

Theorem . Let X be a completeCAT()space such that for any u,v∈X,d(u,v) <π/.

Let S and T be quasinonexpansive and -demiclosed mappings from X into itself with F(S)∩F(T)=∅.Let{αn},{βn}and{γn}be sequences of[a,b]⊂], [.Deﬁne a sequence

{xn} ⊂X by the following recurrence formula:x∈X and ⎧

⎪ ⎨ ⎪ ⎩

un= ( –βn)xn⊕βnSxn,

vn= ( –γn)xn⊕γnTxn,

xn+= ( –αn)un⊕αnvn

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Proof Letz∈F(S)∩F(T). By Corollary ., we have

cosd(un,z)≥( –βn)cosd(xn,z) +βncosd(Sxn,z)

≥( –βn)cosd(xn,z) +βncosd(xn,z) =cosd(xn,z),

cosd(vn,z)≥( –γn)cosd(xn,z) +γncosd(Txn,z)

≥( –γn)cosd(xn,z) +γncosd(xn,z) =cosd(xn,z).

Then, by Corollary . again, we have

cosd(xn+,z)≥( –αn)cosd(un,z) +αncosd(vn,z)

≥( –αn)cosd(xn,z) +αncosd(xn,z)

≥cosd(xn,z).

So, we getd(xn+,z)≤d(xn,z) for alln∈Nand there existsd=limn→∞d(xn,z)≤d(x,z) <

π/.

Furthermore, by Theorem ., we have

cosd(un,z)sind(xn,Sxn)

≥cosd(xn,z)sin( –βn)d(xn,Sxn) +cosd(Sxn,z)sinβnd(xn,Sxn)

≥cosd(xn,z)sin

d(xn,Sxn)

 cos

( – βn)d(xn,Sxn)

 ()

and

cosd(vn,z)sind(xn,Txn)

≥cosd(xn,z)sin( –γn)d(xn,Txn) +cosd(Txn,z)sinγnd(xn,Txn)

≥cosd(xn,z)sin

d(xn,Txn)

 cos

( – γn)d(xn,Txn)

 . ()

Letdn=d(xn,z),sn=d(xn,Sxn)/ andtn=d(xn,Txn)/ forn∈N. If there existsn∈N

such thatsn=tn= , then we havexn∈F(S)∩F(T) and since

xn+= ( –αn)

( –βn)xn⊕βnSxn

⊕αn

( –γn)xn⊕γnTxn

= ( –αn)xn⊕αnxn =xn,

and the proof is ﬁnished. So, we may assumesn=  ortn=  for alln∈N.

Ifsn=  andtn= , then we haveun=xn. From (), (), and Corollary ., we get cosdn+sintncostn

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≥( –αn)cosd(un,z)sintn+αncosd(vn,z)sintn

≥( –αn)cosdnsintncostn+ αncosdnsintncos( – γn)tn.

Dividing by sintn> , we get

cosdn+costn≥( –αn)cosdncostn+αncosdncos( – γn)tn. () Iftn=  andsn= , then we havevn=xn. In a similar way as above, we get

cosdn+cossn≥( –αn)cosdncos( – βn)sn+αncosdncossn. () Ifsn=  andtn= , then from (), (), and Corollary ., we get

cosdn+sinsnsintn

≥( –αn)cosd(un,z)sinsnsintn+αncosd(vn,z)sinsnsintn

≥cosdnsinsnsintn

( –αn)costncos( – βn)sn+αncossncos( – γn)tn

.

Dividing by sinsnsintn> , we get

cosdn+cossncostn

≥( –αn)cosdncostncos( – βn)sn+αncosdncossncos( – γn)tn. ()

Therefore, () and () can be reduced to the inequality () and it is equivalent to

εncossn αncos( – βn)sn

– –αn αn

εncostn ( –αn)cos( – γn)tn

– αn  –αn

≥,

whereεn=cosdn+/cosdnforn∈N. It follows thatlimn→∞εn=cosd/cosd= . Since

{αn} ⊂[a,b]⊂], [ forn∈N, we get

lim inf

n→∞

cossn αncos( – βn)sn

– –αn αn

costn ( –αn)cos( – γn)tn

– αn  –αn

≥. ()

Then we show that there existsn∈Nsuch that for alln≥n, the following hold:

– ≤

– –αn αn ≤

 ()

and

– ≤

– αn  –αn≤

. ()

First, we show the right inequality of (). Since{βn} ⊂[a,b]⊂], [ forn∈N, we get

cossn≤cos| – βn|sn=cos( – βn)sn. Hence we get

– –αn αn

≤ 

αn

– –αn αn

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By the same method as above, the right inequality of () also holds. Next, let us show the left inequality of (). If it does not hold, then letting

σn=

– –αn αn

and τn=

– αn  –αn

,

we can ﬁnd a subsequence{σni} ⊂ {σn}such thatσni< –/ fori∈Nandlimi→∞σni=

σ ≤–/. Since{αn},{γn} ⊂[a,b]⊂], [ and{tn} ⊂[,π/[⊂[,π/[, we have{τn}is bounded. Therefore, by taking a subsequence again if necessary, we may assume that{τni}

converges toτ ∈R. Then, by (), we getσ τ=lim_i_→∞σniτni≥lim infn→∞σnτn≥. Hence

we may assume thatτni<  for alli∈N. Since

√

/ <cossn,

√

/ <costn,  <cos( – βn)sn≤,  <cos( – γn)tn≤ and{αn} ⊂[a,b]⊂], [, we also have

 <

√

 b ≤

and  <

√

 ( –a)≤

. ()

Letρbe a real number such that

 <ρ<min

√

 b,

√

 ( –a),

 –b

b +

a

 –a

. ()

Then, by (), we get

ρ– –αni

αni

≤σni<  and ρ–

αni

 –αni

≤τni< . ()

Then, by () and (), we have

σniτni≤ ρ–

 –αni

αni

ρ– αni

 –αni

=ρ–  –αni

αni

+ αni

 –αni

ρ+ 

≤ρ–  –b

b +

a

 –a

ρ+ 

=ρ ρ–  –b

b +

a

 –a

+ .

Thus, asi→ ∞, we have

≤σ τ≤ρ ρ–  –b

b +

a

 –a

+  < . ()

This is a contradiction. We also obtain the left inequality of () in a similar way. Hence we get

lim

n→∞

– –αn αn

= lim

n→∞

– αn  –αn

=  ()

by Lemma ., (), and (). Furthermore, from (), we get

lim

n→∞

cossn–cos( – βn)sn αncos( – βn)sn

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By Lemma . and (), we get

lim

n→∞

cossn–cos( – βn)sn

= . ()

Moreover, by Lemma . and (), we get

lim inf

n→∞ cossn=lim infn→∞ cos( – βn)sn=lim infn→∞ cos| – βn|sn. Hence we get

lim sup

n→∞ sn=lim supn→∞

| – βn|sn

≤lim sup

n→∞ | – βn|lim supn→∞ sn,

and we have

≥

 –lim sup

n→∞ |  – βn|

lim sup

n→∞

sn=lim inf n→∞

 –| – βn|

lim sup

n→∞

sn.

Since {βn} ⊂[a,b]⊂], [ forn∈N, we have lim infn→∞( –| – βn|) >  and thus,

lim sup_n_→∞sn≤. Therefore, we getlim supn→∞sn=  and we also getlim supn→∞tn= . It impliesd(xn,Sxn)→ andd(xn,Txn)→.

Next, let{yk} be a subsequence of{xn}. Sincer({xn})≤d<π/, by Theorem .(i),

there exists a unique asymptotic centerxof{yk}. Moreover, sincer({yk}) <π/, by The-orem .(ii), there exists a subsequence{zl}of{yk}such thatzl–

z∈X. Further, since d(zl,Szl)→,d(zl,Tzl)→ andS,T are-demiclosed, we havez∈F(S)∩F(T). Then

we can show thatx=z,i.e.,x∈F(S)∩F(T). If not, from the uniqueness of the

asymp-totic centersx,zof{yk},{zl}, respectively, due to Theorem .(i), we have

lim sup

k→∞

d(yk,x) <lim sup

k→∞

d(yk,z)

= lim

n→∞d(xn,z)

=lim sup

l→∞

d(zl,z)

<lim sup

l→∞

d(zl,x)

≤lim sup

k→∞

d(yk,x).

This is a contradiction. Hence we getx∈F(S)∩F(T). Next, we show that for any

sub-sequences of{xn}, their asymptotic center consists of the unique element. Let{uk},{vk} be subsequences of{xn},x∈AC({uk}) andx∈AC({vk}). We showx=xby using

con-tradiction. Assumex=x. Thenx∈/ AC({uk}) andx∈/AC({vk}) by Theorem .(i). It follows that

lim sup

k→∞

d(uk,x) <lim sup

k→∞

duk,x

= lim

n→∞d

xn,x

=lim sup

k→∞ d

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<lim sup

k→∞

d(vk,x)

= lim

n→∞d(xn,x)

=lim sup

k→∞

d(uk,x).

This is a contradiction. Hence we getx=x. Therefore, we have{xn}-converges to a

common ﬁxed point ofSandT.

By Theorem ., we know that a nonexpansive mapping having a ﬁxed point satisﬁes the assumptions in Theorem .. Thus, we get the following result.

Corollary . Let X be a completeCAT()space such that for any u,v∈X,d(u,v) <π/.

Let S and T be nonexpansive mappings of X into itself such that F(S)∩F(T)=∅.Let{αn},

{βn} and{γn}be sequences in[a,b]⊂], [.Deﬁne a sequence{xn} ⊂X as the following

recurrence formula:x∈X and ⎧

⎪ ⎨ ⎪ ⎩

un= ( –βn)xn⊕βnSxn,

vn= ( –γn)xn⊕γnTxn,

for n∈N.Then{xn}-converges to a common ﬁxed point of S and T. 5 An application to the image recovery

The image recovery problem is formulated as to ﬁnd the nearest point in the intersec-tion of family of closed convex subsets from a given point by using corresponding metric projection of each subset. In this section, we consider this problem for two subsets of a completeCAT() space.

Theorem . Let X be a completeCAT()space such that for any u,v∈X,d(u,v) <π/.

Let Cand Cbe nonempty closed convex subsets of X such that C∩C=∅.Let Pand P be metric projections onto Cand C,respectively.Let{αn},{βn}and{γn}be real sequences

in[a,b]⊂], [.Deﬁne a sequence{xn} ⊂X by the following recurrence formula:x∈X and

⎧ ⎪ ⎨ ⎪ ⎩

un= ( –βn)xn⊕βnPxn,

vn= ( –γn)xn⊕γnPxn,

for n∈N.Then{xn}-converges to a ﬁxed point of the intersection of Cand C.

Proof We see thatP andP are quasinonexpansive [] and-demiclosed []. Further,

we also getF(P) =CandF(P) =C. Thus, lettingS=PandT=Pin Theorem ., we

obtain the desired result.

Competing interests

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Authors’ contributions

The authors have contributed in this work on an equal basis. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors thank the anonymous referees for their valuable comments and suggestions. The ﬁrst author is supported by Grant-in-Aid for Scientiﬁc Research No. 22540175 from the Japan Society for the Promotion of Science.

Received: 15 October 2013 Accepted: 23 January 2014 Published:13 Feb 2014 References

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