Coupled fixed point theorems for asymptotically nonexpansive mappings

R E S E A R C H

Open Access

Coupled fixed point theorems for

asymptotically nonexpansive mappings

Hallowed Olaoluwa

_{, Johnson O Olaleru}

_{and Shih-Sen Chang}

*_{Correspondence:} [email protected] 1_{Department of Mathematics,} University of Lagos, Akoka, Lagos, Nigeria

Full list of author information is available at the end of the article

Abstract

We introduce the theory of asymptotical nonexpansiveness of mappings defined in the algebraic productE×Eand with values in the spaceE. We then prove the existence of coupled fixed points of such mappings whenEis a uniformly convex Banach space. This paper is an extension of some recent results in the literature.

MSC: 47H10; 47H09

Keywords: coupled fixed point; nonexpansive map

1 Introduction and preliminaries

In the past years, many researchers have proved various results on the theory of nonex-pansive mappings (or contractions). The mean ergodic theorem for contractions in uni-formly convex Banach spaces was proved in [], while the authors in [] introduced the convex approximation property of a space, proved that contractions satisfy an inequality analogue to the Zarantonello inequality (see []) and then studied the asymptotic behavior of contractions.

Given a nonempty subsetDof a real linear normed spaceE, a self-mappingT:D→D

is said to benonexpansiveif the following inequality holds for allx,y∈D:

Tx–Ty ≤ x–y.

Many more general classes of mappings have been considered, including the class of

asymptotically nonexpansivemappings introduced by Goebel and Kirk [], defined by the relation

Tnx–Tny≤knx–y ∀n≥,∀x,y∈D,

where the sequence{kn} ⊂[, +∞) converges to  asn→+∞. They proved that a self asymptotically nonexpansive map of a nonempty closed convex bounded subset of a real uniformly convex Banach space has a fixed point. Then Changet al.[] established some convergence theorems for this class of mappings without the assumption of boundedness of the subsetD.

Recently, the concept of coupled fixed points was introduced and developed by some authors (see [–]). A coupled fixed point of a mapF:E×E→Eis defined as an element (x,y)∈E×E such thatx=F(x,y) andy=F(y,x). One could be interested in extending

nonexpansiveness to maps defined on a product space (the algebraic product) and study the existence of their coupled fixed points. This is the main purpose of this paper.

We now introduce the definitions of nonexpansive maps, asymptotically nonexpansive maps, Lipschitzian and uniformly Lipschitzian maps defined in product spaces.

Definition . LetDbe a nonempty subset of a real normed linear spaceE. A mapping

F:D×D→Dis said to benonexpansiveif

F(x,y) –F(u,v)≤ 

x–u+y–v ∀x,y,u,v∈X. (.)

Definition . F is said to beasymptotically nonexpansive if there exists a sequence

{kn} ⊂[, +∞) withlimn→+∞kn=  such that

Fn(x,y) –Fn(u,v)≤kn 

x–u+y–v ∀n≥,∀x,y,u,v∈X, (.)

where the sequence{Fn_{}is defined (see []) as follows:}

F(x,y) =x,

Fn+_{(x,y) =F(F}n_(x,y),Fn_{(y,x)), n≥.} (.)

Definition . Fis said to beuniformly L-Lipschitzian(whereLis a positive constant) if

Fn(x,y) –Fn(u,v)≤L 

x–u+y–v ∀n≥,∀x,y,u,v∈X. (.)

When the equality is verified forn= ,i.e., when

F(x,y) –F(u,v)≤L 

x–u+y–v ∀x,y,u,v∈X, (.)

Fis said to beLipschitzwith the constantL(orL-Lipschitzian).

Remark .

. It is easy to see that ifF:D×D→Dis a nonexpansive mapping, thenFis an asymptotically nonexpansive mapping with a constant sequence{}.

. IfT:D×D→Dis an asymptotically nonexpansive mapping with a sequence

{kn} ∈[, +∞)such thatkn→, then it must be uniformlyL-Lipschitzian with

L=sup_n≥{kn}.

. The sequence{Fn_{(x,y)}can be written as the sequence{x}

n}defined (see []) as follows:

⎧ ⎪ ⎨ ⎪ ⎩

x=x, y=y, xn+=F(xn,yn), n≥,

yn+=F(yn,xn), n≥.

(.)

In [], Changet al.defined demi-closed maps at the origin as follows.

Definition .[] LetEbe a real Banach space andDbe a closed subset ofE. A

We extend this definition to maps defined inD×Das follows.

Definition . LetEbe a real Banach space andDbe a closed subset ofE. A mapping

F :D×D→Dis said to be demi-closed at the originif, for any sequence{(xn,yn)} in

D×D, the conditionsxn→q,yn→qweakly andF(xn,yn)→,F(yn,xn)→ strongly implyF(q,q) =F(q,q) = .

Lemma . Let E be a uniformly convex Banach space,C be a nonempty bounded closed

convex subset of E. Then there exists a strictly increasing continuous convex function f : [, +∞)→[, +∞)with f() = such that for any Lipschitzian mapping T:C×C→E with the Lipschitz constant L≥, any finite many elements{(xi,yi)}ni= in C ×C and any finite many nonnegative numbers {ti}ni= with

n

i=ti = , the following inequality

holds:

T

_n

i= ti(xi,yi)

– n

i=

tiT(xi,yi)

≤L

f

–

max ≤i,j≤n

xi–xj+yi–yj–

L



–

T(xi,yi) –T(xj,yj)

. (.)

Proof Let us prove by induction. Forn= , (.) is trivial.

Let us prove forn= : Letδbe the modulus of uniform convexity ofEand defined: R+_→R+_by

d(t) =

 

t

δ(s)ds, ≤t≤, d() +

δ()(t– ), t> .

It is well known (e.g., see [, ]) thatdis strictly increasing, continuous, convex, satisfying

d() =  and

ttd

u–v≤ –tu+tv (.)

for allt,t≥ such thatt+t=  andu,v∈Esuch thatu ≤ andv ≤.

T(x,y) –T

_

i= ti(xi,yi)

=T(x,y) –T(tx+tx,ty+ty) ≤L



x–tx–tx+y–ty–ty

=L

( –t)x–tx+( –t)y–ty =L



tx–tx+ty–ty

=L t

x–x+y–y

Similarly,

T(x,y) –T

_

i=

ti(xi,yi)

=T(x,y) –T(tx+tx,ty+ty) ≤L



x–tx–tx+y–ty–ty

=L

( –t)x–tx+( –t)y–ty =L



tx–tx+ty–ty

=L t

x–x+y–y

. (.)

Hence, ifu=T(x,y)–T( 

i=ti(xi,yi)) L

t[x–x+y–y]

andv=T( 

i=ti(xi,yi))–T(x,y) L

t[x–x+y–y] , then from (.) and (.),

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

u ≤, v ≤,

tu+tv= LT(x,y)–T(x,y) [x–x+y–y], u–v=tT(x,y)+tT(x,y)–T(

 i=ti(xi,yi)) L

tt[x–x+y–y] .

(.)

Putting (.) in (.), we have

ttd

tT(x,y) +tT(x,y) –T(



i=ti(xi,yi)) L

tt[x–x+y–y]

≤ – _LT(x,y) –T(x,y)

[x–x+y–y]

,

i.e.,

ttL

x–x+y–y

d

tT(x,y) +tT(x,y) –T(



i=ti(xi,yi)) L

tt[x–x+y–y]

≤L



x–x+y–y

–T(x,y) –T(x,y).

Sincet→d(_tt) is strictly increasing,tt≤ _ andx–x+y–y ≤D, whereD:= diamC, we have

L

Dd

tT(x,y) +tT(x,y) –T(



i=ti(xi,yi)) L

D

≤ttL

x–x+y–y

d

tT(x,y) +tT(x,y) –T(



i=ti(xi,yi)) L

tt[x–x+y–y]

≤L



x–x+y–y

–T(x,y) –T(x,y).

Letf:R+→R+be defined asf(t) =Dd(_Dt); then

L

f



L

tT(x,y) +tT(x,y) –T

_

i=

ti(xi,yi)

≤L



x–x+y–y

Hence,

tT(x,y) +tT(x,y) –T

_

i= ti(xi,yi)

≤L f – 

x–x+y–y–



LT(x,y) –T(x,y)

. (.)

Thus (.) is true forn= . Now suppose that

T _n i= ti(xi,yi)

– n

i=

tiT(xi,yi)

≤L f – n max ≤i,j≤n

xi–xj+yi–yj–

L



–

T(xi,yi) –T(xj,yj)

. (.)

Define a strictly increasing continuous convex functionfn+satisfyingfn+() =  and

f_n–_+(t)≥f_–(t) +f_n–t+ f_–(t).

For example, if∗defines the inf-convolution,Ithe identity function onRand the functions in the functionsfnare extended to be +∞on (–∞, ), then one could take (see [])

fn+=

 (I∗f)o

 (f∗fn)

.

Define alson–_{={t= (t}

, . . . ,tn) :ti≥,

n i=ti= }. Fixt∈n_{and let (x}

,y), . . . , (xn+,yn+)∈C×C. The casetn+=  is trivial and therefore

omitted. For the rest of the proof, we let the subscriptirange through ≤i≤n+ , while

jranges through ≤j≤n. Put

uj

vj

= ( –tn+)

xj

yj

+tn+

xn+ yn+

,

x_i y_i

=

T(xi,yi)

T(yi,xi)

,

u_j v_j

= ( –tn+)

x_j y_j

+tn+

x_n+ y_n+

,

μj=

tj  –tn+

.

We have thatμ∈n–_{and after computation}

n+ i= ti xi yi = n j= μj uj vj , n+ i= ti

x_i y_i

= n j= μj

u_j v_j

T _n+ i= ti(xi,yi)

– n+

i= tixi

= T _n j=

μj(uj,vj)

– n

j=

μjuj

≤T

_n

j=

μj(uj,vj)

– n

j=

μjT(uj,vj)

+ n j=

μjT(uj,vj) –uj. (.)

The induction hypothesis (.) can also be written

fn  L T _n j=

μj(uj,vj)

– n

j=

μjT(uj,vj)

≤ max

≤j,k≤n

uj–uk+vj–vk– 

LT(uj,vj) –T(uk,vk)

= max ≤j,k≤n

uj–uk+vj–vk– 

LT(vj,uj) –T(vk,uk)

= max ≤j,k≤n

uj–uk+vj–vk– 

LT(uj,vj) –T(uk,vk)

– 

LT(vj,uj) –T(vk,uk)

. (.)

We also have, by the triangle inequality,

uj–uk– 

LT(uj,vj) –T(uk,vk) ≤ uj–uk–



Lu

j–uk+ 

Lu

k–T(uk,vk)+ 

Lu

j–T(uj,vj), (.)

vj–vk– 

LT(vj,uj) –T(vk,uk) ≤ vj–vk–



Lv

j–vk+ 

Lv

k–T(vk,uk)+ 

Lv

j–T(vj,uj). (.) From (.) we have

f



LT(uj,vj) –u

j

=f



LT(uj,vj) – ( –tn+)x

j–tn+xn+

=f



LT(uj,vj) – ( –tn+)T(xj,yj) –tn+T(xn+,yn+)

=f



LT

( –tn+)xj+tn+xn+, ( –tn+)yj+tn+yn+

– ( –tn+)T(xj,yj) –tn+T(xn+,yn+)

≤ xj–xn++yj–yn+–



LT(xj,yj) –T(xn+,yn+)

=xj–xn++yj–yn+–



Lx

Similarly,

f



LT(vj,uj) –v

j

≤ xj–xn++yj–yn+– 

Ly

j–yn+. (.)

We also have

uj–uk– 

Lu

j–uk=( –tn+)(xj–xk)– 

L( –tn+)

x_j–x_k

= ( –tn+)

xj–xk– 

Lx

j–xk

≤ xj–xk– 

Lx

j–xk. (.) Similarly,

vj–vk– 

Lv

j–vk≤ yj–yk– 

Ly

j–yk. (.) Put

t:=max

xi–xk+yi–yk– 

Lx

i–xk: ≤i,k≤n+ 

=max

xi–xk+yi–yk– 

Ly

i–yk: ≤i,k≤n+ 

=max

xi–xk+yi–yk– 

Lx

i–xk– 

Ly

i–yk: ≤i,k≤n+ 

.

Then, by (.) and (.),



LT(uj,vj) –u

j≤f–(t) and



LT(vj,uj) –v

j≤f–(t). (.)

Using (.) and (.) in (.) and also (.) and (.) in (.), we have

uj–uk– 

LT(uj,vj) –T(uk,vk)≤ xj–xk–



Lx

j–xk+f–(t)

and

vj–vk– 

LT(vj,uj) –T(vk,uk)≤ yj–yk–



Ly

j–yk+f–(t);

by summing,

uj–uk+vj–vk– 

LT(uj,vj) –T(uk,vk)–



LT(vj,uj) –T(vk,uk) ≤ xj–xk+yj–yk–



Lx

j–xk– 

Ly

j–yk+ f–(t)

Using (.) in (.) yields

T

_n

j=

μj(uj,vj)

– n

j=

μjT(uj,vj)

≤

L

f

–

n

t+ f_–(t). (.)

Finally, when (.) and (.) are used in (.), we obtain

T

_n+

i= ti(xi,yi)

– n+

i=

tiT(xi,yi)

≤

L

f

–

n

t+ f_–(t)+ n

j=

μj

L

f

–  (t)

≤L

f

–

n

t+ f_–(t)+L f

–  (t)

=L 

f_n–t+ f_–(t)+f_–(t)

≤L

f

–

n+(t) by the definition offn+.

Therefore

T

_n+

i= ti(xi,yi)

– n+

i=

tiT(xi,yi)

≤L

f

–

n+

max ≤i,k≤n+

xi–xk+yi–yk– 

LT(xi,yi) –T(xk,yk)

.

To complete the proof, we show in the sequel that the dependence offnonncan actually be omitted.

SinceEis uniformly convex,EisB-convex (see []) and since the product ofB-convex spaces is alsoB-convex (see []),E _{isB-convex, hence has the convex approximation}

property (C.A.P.) (see []),i.e., for each> , there exists a positive integerpsuch that

coM⊂copM+S

, L

×S

, L

×S

, L

,

whereS(,r) is the open sphere centered at the origin and withras radius,coMis the convex hall ofMand

copM=

_p

i=

tiXi;t∈p–;Xi∈Mfor alli∈ {, . . . ,p},pfixed

for everyM⊂C×C×C.

Putγ =fp(_L). Supposex, . . . ,xn,y, . . . ,yn∈Csatisfy

xi–xj+yi–yj– 

LT(xi,yi) –T(xj,yj)≤γ for alli,j.

μ∈p–_andi

, . . . ,ip∈ {, . . . ,n}such that

n i= tixi–

p

j=

μjxij

<

L,

n i= tiyi–

p

j=

μjyij

<

L,

n i=

tiT(xi,yi) – p

j=

μjT(xij,yij)

<

L≤

.

Now

Tti(xi,yi)

–tiT(xi,yi)≤T

ti(xi,yi)

–Tμj(xij,yij)

+Tμj(xij,yij)

–μjT(xij,yij)

+μjT(xij,yij) –

tiT(xi,yi). Since

Tti(xi,yi)

–Tμj(xij,yij)

≤L



tixi–

μjxij+

tiyi–

μjyij

<L 

L+

L =  and

Tμj(xij,yij)

–μjT(xij,yij)

≤L

f

–

p (γ) =

L f – p fp  L = ,

we have that

Tti(xi,yi)

–tiT(xi,yi)<

+ + =

wheneverxi–xj+yi–yj–_LT(xi,yi) –T(xj,yj) ≤γ. One can easily constructf such that < L_f–_{(γ),i.e., such thatf(t)≤f}

p(_t), which guarantees (.) withf independent

ofn.

2 Existence of coupled fixed points

Letp:E×E→E denote the projection of the first coordinate,i.e.,p(x,y) =xfor all x,y∈X. Note that ifF:D×D→Dis a mapping, we have thatp–Fis demi-closed at the

origin if the existence of a sequence{(xn,yn)}inD×Dsuch that

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

xn→q weakly, yn→q weakly,

xn–F(xn,yn)→ strongly,

implies that

(p–F)(q,q) = (p–F)(q,q) = ,

i.e.,

F(q,q) =q and q=F(q,q),

i.e., the existence of a coupled fixed point (q,q) ofF.

Now we prove our main theorem.

Theorem . Let D be a nonempty closed convex subset of a uniformly convex Banach

space E and F:D×D→D be an asymptotically nonexpansive map with the sequence {kn}as defined in(.).Then p–F satisfies the demi-closedness at the origin property.In other terms,if any sequence{(xn,yn)}in D×D is such that xn→qweakly,yn→qweakly, xn–F(xn,yn)→strongly and yn–F(yn,xn)→strongly,then F has a coupled fixed point (q,q).

Proof Since{(xn,yn)}converges weakly toq= (q,q)∈D×D,{xn}and{yn}are bounded inD. Therefore, there existsr>  such that{xn},{yn} ⊂C=:D∩B[,r], whereB[,r] is the ball ofEof radiusrcentered in . HenceCis a nonempty bounded closed convex subset inD.

Next, we prove that asn→+∞,Fnq→q andFnq¯→q, whereq= (q,q) andq¯=

(q,q).

Since{xn}and{yn}converge weakly toqandqrespectively, by Mazur’s theorem (see, e.g., []), for alln≥, there exist sequences{An}and{Bn}such thatAn=

m(n)

i= t (n)

i xi+nand

Bn=

m(n)

i= t (n)

i yi+n, wheret(in)≥,

m(n)

i= t (n)

i =  andAn–q<_n,Bn–q<_n.

Since the sequences{xn–F(xn,yn)}and{yn–F(yn,xn)}converge strongly to  respec-tively, for any given>  and positive integerj≥, there is an integerN=N(,j) such that



N <and

_{xn–F(xn,yn)+yn–F(yn,xn)≤}   +j_l–_=kl/

<, n≥N.

SinceFis asymptotically nonexpansive,

Fj(xn,yn) –Fj+(xn,yn)=Fj(xn,yn) –Fj

F(xn,yn),F(yn,xn)

≤kj

xn–F(xn,yn)+yn–F(yn,xn) and

Fj(yn,xn) –Fj+(yn,xn)=Fj(yn,xn) –Fj

F(yx,yn),F(xn,yn)

≤kj

Hence, for anyn≥N,

xn–Fj(xn,yn)≤xn–F(xn,yn)+F–F

(xn,yn)+· · · +Fj––Fj(xn,yn)

≤

 + j–

l= kl/

_{xn–F(xn,yn)}+yn–F(yn,xn)

<. (.)

Similarly,

yn–Fj(yn,xn)<. (.)

SinceF:D×D→Dis asymptotically nonexpansive,F:C×C→Dis also asymptotically nonexpansive, henceFj_{:C×C→Dis a Lipschitzian mapping with the Lipschitz constant}

kj≥. We have the following inequality:

_Fj_(A

n,Bn) –An=

Fj(An,Bn) – m(n)

i=

t(_in)Fj(xi+n,yi+n)

+ m(n)

i=

t_i(n)Fj(xi+n,yi+n) – m(n)

i= t_i(n)xi+n

≤Fj(An,Bn) – m(n)

i=

t(_in)Fj(xi+n,yi+n)

+ m(n)

i=

t_i(n)Fj(xi+n,yi+n) –xi+n. (.)

By (.), we know that

m(n)

i=

t(_in)Fj(xi+n,yi+n) –xi+n< ∀n≥N. (.)

By Lemma ., (.) and (.), we have

Fj(An,Bn) – m(n)

i=

t(_in)Fj(xi+n,yi+n)

≤(kj/)f–

max ≤i,k≤m(n)

xi+n–xk+n+yi+n–yk+n

– (kj/)–Fj(xi+n,yi+n) –Fj(xk+n,yk+n)

≤(kj/)f–

max ≤i,k≤m(n)

_xi+n–Fj_(x

– (kj/)–Fj(xi+n,yi+n) –Fj(xk+n,yk+n)

≤(kj/)f–

max ≤i,k≤m(n)

+ –k–_j kj

xi+n–xk+n+yi+n–yk+n

≤(kj/)f–

+ r –k_j–kj

, n≥N. (.)

Inequalities (.) and (.) into (.) yield

Fj(An,Bn) –An≤(kj/)f–

+ rkj

 –k_j–+.

Taking the limit superior asn→+∞in the above inequality and noting that>  is arbi-trary, we have

lim sup

n→+∞

Fj(An,Bn) –An≤(kj/)f–

rkj

 –k_j–. (.)

Also, by the definition of the sequences{An}and{Bn}, we have that for allj≥,

Fj(q) –q≤Fj(q,q) –Fj(An,Bn)+Fj(An,Bn) –An+An–q

≤(kj+ )

An–q+Bn–q

+Fj(An,Bn) –An

≤ 

n(kj+ ) +F

j_(A

n,Bn) –An.

Taking the limit superior in the above inequality, from (.) we have

Fj(q) –q≤(kj/)f–

rkj

 –k_j–.

Finally, the limit superior in the above inequality yieldslim sup_j→+∞Fj_{(q) –q ≤f}–_{() =}

, which implies thatFj(q) –q → asj→+∞.

Similar computations yield thatFj_{(q¯) –q → asj→+∞. Hence,}

F(qj,qj)→q, F(qj,qj)→q.

SinceFis continuous, we have

q=limj→+∞F(qj+,qj+) =limj→+∞F(F(qj,qj),F(qj,qj)) =F(q,q), q=limj→+∞F(qj+,qj+) =limj→+∞F(F(qj,qj),F(qj,qj)) =F(q,q).

Henceq= (q,q) is a coupled fixed point ofF, which completes the proof.

The conclusion of demi-closedness in the previous theorem states that if sequences{xn} and{yn}inDare such thatxn→q,yn→qweakly andxn–F(xn,yn)→,yn–F(yn,xn)→  strongly, then (q,q) is a coupled fixed point ofF. A more direct conclusion about the

Theorem . Let D be a nonempty convex closed and bounded subset of a uniformly con-vex Banach space E.Then any asymptotically nonexpansive mapping T:D×D→D has a coupled fixed point.

Proof Letu,v∈Dbe fixed. Define the setR(u,v) as follows:

R(u,v) =

ρ∈R/∃kρ∈N:D∩

_+∞

i=kρ

SFi(u,v) +Fi(v,u),ρ

=∅

,

whereS(x,r) is the open sphere inEof centerxand radiusr.Dis bounded, so ifd:=diamD

(diameter ofD),d∈R(u,v), henceR(u,v)=∅. Letρ*be the g.l.b. ofR(u,v). For each> , define the set

A:=

+∞

!

k=

_+∞

i=k

SFi(u,v) +Fi(v,u),ρ*+

.

The sets A ∩D are nonempty and convex. Since E is reflexive, A:=

"

>A ∩Dis nonempty. Now, for anyx,y∈Aandη> , there existsN∈Nsuch thati≥N implies thatx–Fi(u,v)+y–Fi(v,u) ≤ρ*+η.

Let us show that there existx,y∈Asuch that the sequences{Fn_(x,y)}and{Fn_(y,x)} con-verge toxandyrespectively. Suppose that for allx,y∈Asuch sequences do not converge. Then∃>  and a strictly increasing sequence of integers{ni}is such that

Fni_{(x,y) –x+F}ni_{(y,x) –y}≥_{, i= , , . . . .}

Form>n,

Fn(x,y) –Fm(y,x)=Fn(x,y) –FnFm–n(x,y),Fm–n(y,x)

≤kn

x–Fm–n_(x,y)+y–Fm–n_(y,x)

 ,

Fn(y,x) –Fm(y,x)=Fn(y,x) –FnFm–n(y,x),Fm–n(y,x)

≤kn

x–Fm–n_(x,y)+y–Fm–n_(y,x)

 .

Hence

Fn(y,x) –Fm(y,x)+Fn(y,x) –Fm(y,x)

≤knx–Fm–n(x,y)+y–Fm–n(y,x).

Letδbe the modulus of convexity of the spaceE. Assumeρ*>  andα>  so that ( –

δ(_ρ*_+α))(ρ*+α) <ρ*, and selectnso thatx–Fn(x,y)+y–Fn(y,x) ≥ and so that kn(ρ*+α_)≤ρ*+α.

IfN≥nis sufficiently large, thenm>Nimplies that

and also

Fn(x,y) –Fm(u,v)+Fn(y,x) –Fm(v,u)≤knx–Fm–n(u,v)

+y–Fm–n(v,u)

≤kn

ρ*+α 

≤ρ*+α.

Now, by the uniform convexity ofE,

X+Y



≤

 –δ

ρ*_+α

ρ*_+α, forX,Y ≤ρ

*_{+α,X–Y ≥.}

Sincex–Fm_(u,v)+y–Fm_{(v,u) ≤ρ}*₊α

, lettingX=x–Fm(u,v) +y–Fm(v,u) and Y=Fn_{(x,y) +F}n_{(y,x) –F}m_{(u,v) –F}m_{(v,u), we have ifm>N}

x+Fn(x,y) +_y+Fn(y,x)–Fm(u,v) +Fm(v,u)≤

 –δ

ρ*_+α

ρ*+α<ρ*.

This contradicts the definition ofρ*_{, the g.l.b. ofR(u,v), since}

ρ*>

 –δ

ρ*_+α

ρ*+α∈R(u,v).

Hence,ρ*_{=  and sox=F(x,y) andy=F(x,y). The fact thatρ}*_{=  implies that the}

se-quences{Fn_(u,v)}and{Fn_{(v,u)}are Cauchy, henceF}n_{(u,v)→x=F(x,y) andF}n_(v,u)→

y=F(y,x). This completes the proof.

Since nonexpansive maps are asymptotically nonexpansive, we have the following corol-lary.

Corollary . Let D be a nonempty convex closed and bounded subset of a uniformly

con-vex Banach space.Then any nonexpansive mapping T:D×D→D has a coupled fixed point.

Remark . Theorem . is an extension of Theorem  of [] to nonexpansive maps

de-fined in a product space. The proof of Theorem . follows the methodology in [], ex-tending the result therein to product spaces. Our results are, to the best of our knowledge, first of their kind in the theory of nonexpansiveness in product spaces dealing with the existence of a coupled fixed point.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Author details

1_{Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria.}2_{College of Statistics and Mathematics, Yunnan} University of Finance and Economics, Kunming, Yunnan 650221, China.

Received: 21 September 2012 Accepted: 1 February 2013 Published: 25 March 2013

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doi:10.1186/1687-1812-2013-68