F-cone metric spaces over Banach algebra

R E S E A R C H

Open Access

F

-cone metric spaces over Banach algebra

Jerolina Fernandez

_{, Neeraj Malviya}

_{, Stojan Radenoviˇc}

_{and Kalpana Saxena}

*_{Correspondence:}

[email protected]

2_{Nonlinear Analysis Research Group,}

Ton Duc Thang University, Ho Chi Minh City, Vietnam

3_{Faculty of Mathematics and}

Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Full list of author information is available at the end of the article

Abstract

The paper deals with the achievement of introducing the notion ofF-cone metric spaces over Banach algebra as a generalization ofNp-cone metric space over Banach algebra andNb-cone metric space over Banach algebra and studying some of its topological properties. Also, here we define generalized Lipschitz and expansive maps for such spaces. Moreover, we investigate some fixed points for mappings satisfying such conditions in the new framework. Subsequently, as an application of our results, we provide an example. Our results generalize some well-known results in the literature.

MSC: 46B20; 46B40; 46J10; 54A05; 47H10

Keywords: F-cone metric space over Banach algebra;c-sequence; generalized Lipschitz mapping; expansive mapping; fixed point

1 Introduction

Partial metric spaces were introduced by Matthews [] in . He studied a partial metric space as a part of the denotational semantics of dataflow networks and showed that the Banach contraction principle can be generalized to the partial metric context for appli-cations in program verification. Especially, it has the property that differentiates it from other spaces, that is, the self-distance of any point may not be zero, also a convergent sequence need not have unique limit in these spaces.

On the other hand, in , the concept ofb-metric spaces was introduced by Bakhtin [] as a generalization of metric spaces. He showed the contraction mapping principle in ab-metric space that generalizes the prominent Banach contraction principle in metric spaces.

In the same spirit, recently, Huang and Zhang [] replaced the set of real numbers by ordering Banach space and defined a cone metric space as a generalization of the metric space. The authors proved some fixed point theorems of contractive mappings on cone metric spaces. They also defined the Cauchy sequence and convergence of a sequence in such spaces in terms of interior points of the underlying cone. After that, in [], Rezapour and Hamlbarani generalized some results of [] by omitting the assumption of normal-ity. For fixed point theorems on cone metric spaces, readers may refer to [–] and the references therein.

Malviya et al. [] introduced the concept ofN-cone metric spaces, which is a new gen-eralization of the generalizedG-cone metric space [] and the generalized D∗-metric space []. They proved fixed point theorems for asymptotically regular maps and

quences. Afterwards, in [], the authors defined contractive maps in N-cone metric spaces and proved various fixed point theorems for such maps.

Despite these features, some authors demonstrated that the fixed point results proved on cone metric spaces are the straightforward outcome of the corresponding results of usual metric spaces where the real-valued metric functiond∗ is defined by a nonlinear scalarization functionξe(see []) or by a Minkowski functionalqe(see []).

Due to the concrete reasons mentioned above, researchers were losing their interest in a cone metric space. Fortunately, Liu and Xu [] introduced the approach of cone metric spaces over Banach algebras by replacing Banach spacesEby Banach algebrasAas the underlying spaces of cone metric spaces. They verified some fixed point theorems of gen-eralized Lipschitz mappings with weaker conditions on the gengen-eralized Lipschitz constant

kby means of the spectral radius. Not long ago, Xu and Radenović [] deleted the nor-mality of cones and greatly generalized the main results of []. In particular, some authors have shown recently some fixed point results given in [–].

Following these ideas, very recently, Fernandez et al. [] introduced the notion of partial cone metric spaces over Banach algebra as a generalization of partial metric spaces and cone metric spaces over Banach algebra, which was selected for Young Scientist Award , M.P., India (see []).

Recently, proceeding in this direction, Fernandez et al. introduced the structure ofN p-cone metric space over Banach algebra [] as a generalization ofN-cone metric space over Banach algebra [] and partial metric space andNb-cone metric space over Banach algebra [] as a generalization ofN-cone metric space over Banach algebra [] andb -metric space, respectively.

Inspired and encouraged by the previous works, we present seven sections in this paper. For the reader’s convenience, we recall in Section  some definitions that will be used in the sequel. In Section , after the preliminaries, we introduceF-cone metric spaces over Banach algebra which generalizeNp-cone metric spaces over Banach algebra andNb-cone metric spaces over Banach algebra. In Section , we discuss the topological properties. In Section , we introduce the notions of generalized Lipschitz and expansive maps. Section  is devoted to deriving the existence of fixed point theorems for such spaces by using the mentioned contractive conditions. Finally, in Section , we define expansive maps and in-vestigate the existence and uniqueness of the fixed point in the new framework. Our main theorems extend and unify existing results in the recent literature. We also give illustrative examples that verify our results.

2 Preliminaries

We begin with the following definition as a recall from [].

LetAalways be a real Banach algebra. That is,Ais a real Banach space in which an operation of multiplication is defined subject to the following properties (for allx,y,z∈A,

α∈R):

. (xy)z=x(yz);

. x(y+z) =xy+xzand(x+y)z=xz+yz; . α(xy) = (αx)y=x(αy);

. xy ≤ xy.

invertible if there is an inverse element y∈Asuch thatxy=yx=e. The inverse ofxis denoted byx–_{. For more details, we refer the reader to [].}

The following proposition is given in [].

Proposition . Let A be a Banach algebra with a unit e,and x∈A.If the spectral radius

ρ(x)of x is less than,i.e.,

ρ(x) = lim

n→∞x

nn_=infxnn_{< ,}

then e–x is invertible.Actually,

(e–x)–= ∞

i= xi.

Remark . From [] we see that the spectral radiusρ(x) ofxsatisfiesρ(x)≤ xfor all

x∈A, whereAis a Banach algebra with a unite.

Remark .(see []) In Proposition ., if the conditionρ(x) < is replaced byx ≤, then the conclusion remains true.

Remark .(see []) Ifρ(x) < , thenxn_{→ (n→ ∞).}

Lemma .(see []) If E is a real Banach space with a solid cone P and ifθu c for

eachθ c,then u=θ.

A subsetPofAis called a cone ofAif . Pis nonempty closed and{θ,e} ⊂P;

. αP+βP⊂Pfor all nonnegative real numbersα,β; . P_=PP⊂P;

. P∩(–P) ={θ},

whereθdenotes the null of the Banach algebraA. For a given coneP⊂A, we can define a partial orderingwith respect toPbyxyif and only ify–x∈P.x≺ywill stand for

xyandx=y, whilex ywill stand fory–x∈intP, whereintPdenotes the interior ofP. IfintP=∅, thenPis called a solid cone.

The conePis called normal if there is a numberM>  such that, for allx,y∈A,

θxy ⇒ x ≤My.

The least positive number satisfying the above is called the normal constant ofP[]. In the following we always assume thatAis a Banach algebra with a unite,Pis a solid cone inAandis the partial ordering with respect toP.

Definition .([, ]) LetXbe a nonempty set. Suppose that the mappingd:X×X→A

satisfies

. θ≺d(x,y)for allx,y∈Xandd(x,y) =θ if and only ifx=y; . d(x,y) =d(y,x)for allx,y∈X;

Thendis called a cone metric onX, and (X,d) is called a cone metric space over Banach algebraA.

For other definitions and related results on cone metric space over Banach algebra, we refer to [].

Definition .([]) LetXbe a nonempty set ands≥ be a given real number. A function

d:X×X→R+is ab-metric onXif, for allx,y,z∈X, the following conditions hold: . d(x,y) = if and only ifx=y;

. d(x,y) =d(y,x);

. d(x,z)≤s[d(x,y) +d(y,z)].

In this case, the pair (X,d) is called ab-metric space.

For more definitions and results onb-metric spaces, we refer the reader to [].

Definition .([]) A partial metric on a nonempty setXis a functionp:X×X→R+

such that for allx,y,z∈X, the following conditions hold: . x=y⇔p(x,x) =p(x,y) =p(y,y);

. p(x,x)≤p(x,y); . p(x,y) =p(y,x);

. p(x,y)≤p(x,z) +p(z,y) –p(z,z).

The pair (X,p) is called a partial metric space. It is clear that ifp(x,y) = , then from () and ()x=y. But ifx=y,p(x,y) may not be .

Definition .([]) LetXbe a nonempty set. A functionN:X_{→Ais calledN-cone}

metric onXif for anyx,y,z,a∈X, the following conditions hold:

(N) ≤N(x,x,x);

(N) N(x,y,z) =θ iffx=y=z;

(N) N(x,y,z)≤N(x,x,a) +N(y,y,a) +N(z,z,a).

Then the pair (X,N) is called anN-cone metric space over Banach algebraA.

Definition .([]) AnNb-cone metric on a nonempty setXis a functionNb:X→A

such that for allx,y,z,a∈X:

(Nb) θNb(x,y,z);

(Nb) Nb(x,y,z) =θ iffx=y=z;

(Nb) Nb(x,y,z)s[Nb(x,x,a) +Nb(y,y,a) +Nb(z,z,a)].

The pair (X,Nb) is called anNb-cone metric space over Banach algebraA. The number

s≥ is called the coefficient of (X,Nb).

Definition .([]) AnNp-cone metric on a nonempty setXis a functionNp:X→A

such that for allx,y,z,a∈X:

(Np) x=y=z⇔Np(x,x,x) =Np(y,y,y) =Np(z,z,z) =Np(x,y,z);

(Np) θNp(x,x,x)Np(x,x,y)Np(x,y,z), for allx,y,z∈Xwithx=y=z; (Np) Np(x,y,z)Np(x,x,a) +Np(y,y,a) +Np(z,z,a) –Np(a,a,a).

3 F-cone metric space over Banach algebra

In this section, we define a new structure, i.e.,F-cone metric spaces over Banach algebra.

Definition . LetXbe a nonempty set. A functionF:X_{→Ais calledF-cone metric}

onXif for anyx,y,z,a∈X, the following conditions hold:

(F) x=y=ziffF(x,x,x) =F(y,y,y) =F(z,z,z) =F(x,y,z);

(F) θF(x,x,x)F(x,x,y)F(x,y,z), for allx,y,z∈Xwithx=y=z;

(F) F(x,y,z)s[F(x,x,a) +F(y,y,a) +F(z,z,a)] –F(a,a,a).

Then the pair (X,F) is called anF-cone metric space over Banach algebraA. The number

s≥ is called the coefficient of (X,F).

Remark . In an F-cone metric space over Banach algebra (X,F), if x,y,z ∈X and

F(x,y,z) =θ, thenx=y=z, but the converse may not be true.

Example . LetA=CR

[, ] and define a norm onAbyx=x∞+x∞ forx∈A.

Define multiplication inAas just pointwise multiplication. ThenAis a real unit Banach algebra with unite= . SetP={x∈A:x≥}is a cone inA. Moreover,Pis not normal (see []). LetX= [,∞). Define a mappingF:X→Aby F(x,y,z)(t) = ((max{x,z})+ (max{y,z})_{, (max{x,z})}_{+ (max{y,z})}_)et _{for allx,y,z∈X, and letα>  be any constant.} Then (X,F) is anF-cone metric space over Banach algebraAwith the coefficients= . But it is not anNp-cone metric space over Banach algebra since the triangle inequality is not satisfied; neither it is anNb-cone metric space over Banach algebraAsince for anyx> , we haveNb(x,x,x)(t) = x_.et_=θ.

Lemma . Let(X,F)be an F-cone metric space over Banach algebra A.Then

(a) ifF(x,y,z) =θ,thenx=y=z. (b) ifx=y,thenF(x,x,y) >θ.

Proof The proof is obvious.

Proposition . If (X,F) is an F-cone metric space over Banach algebra, then for all

x,y,z∈X,we have F(x,x,y) =F(y,y,x).

Definition . Let (X,F) be anF-cone metric space over Banach algebra A. Then, for

x∈Xandc>θ, theF-balls with centerxand radiusc>θare

BF(x,c) =

y∈X:F(x,x,y) F(x,x,x) +c.

4 Topology onF-cone metric space over Banach algebra

In this section, we define the topology ofF-cone metric space over Banach algebra and study its topological properties.

Definition . Let (X,F) be anF-cone metric space over Banach algebraAwith

Remark . Let (X,F) be anF-cone metric space over Banach algebraA. In this paper,τ

denotes the topology onX,Bdenotes a subbase for the topology onτandBF(x,c) denotes theF-ball in (X,F), which are described in Definition .. In addition,Udenotes the base generated by the subbaseB.

Theorem . Let(X,F)be an F-cone metric space over Banach algebra A,and let P be a

solid cone in A.Let k∈P be an arbitrarily given vector,then(X,F)is a Hausdorff space.

Proof Let (X,F) be anF-cone metric space over Banach algebra, and letx,y∈Xwithx=y. LetF(x,x,y) =c.

SupposeU=B(x,_c) andV=B(y,_c). Thenx∈Uandy∈V.

We claim thatU∩V=φ. If not, there existsz∈U∩V. But then

F(x,x,z)≺ c

s and F(y,y,z)≺ c

s.

So, we get

c=F(x,x,y)sF(x,x,z) +F(x,x,z) +F(y,y,z)–F(z,z,z)

≺sF(x,x,z) +F(y,y,z)

s

c

s+ c

s

≺c,

i.e.,c<c, which is a contradiction.

HenceU∩V=φandXis a Hausdorff space.

Now, we defineθ-Cauchy sequence and convergent sequence in anF-cone metric space

over Banach algebraA.

Definition . Let (X,F) be anF-cone metric space over Banach algebraA. A sequence

{xn}in (X,F) converges to a pointx∈Xwhenever for everycθthere is a natural number

Nsuch thatF(xn,x,x) cfor alln≥N. We denote this by

lim

n→∞xn=x or xn→x (n→ ∞).

Definition . Let (X,F) be anF-cone metric space over Banach algebraA. A sequence

{xn}inXis said to be aθ-Cauchy sequence in (X,F) if{F(xn,xm,xm)}is ac-sequence inA, i.e., if for everycθ there existsn∈Nsuch thatF(xn,xm,xm) cfor alln,m≥n.

Definition . Let (X,F) be anF-cone metric space over Banach algebraA. ThenXis

Definition . Let (X,F) and (X,F) be anF-cone metric space over Banach algebraA. Then a function f :X→Xis said to be continuous at a pointx∈X if and only if it is sequentially continuous atx, that is, whenever{xn}is convergent tox, we have{fxn}is convergent tof(x).

5 Generalized Lipschitz maps

In this section, we define generalized Lipschitz maps inF-cone metric spaces over Banach algebra.

Definition . Let (X,F) be anF-cone metric space over Banach algebraAandPbe a

cone inA. A mapT:X→Xis said to be a generalized Lipschitz mapping if there exists a vectork∈Pwithρ(k) <  for allx,y∈Xsuch that

F(Tx,Tx,Ty)kF(x,x,y).

Example . Let the Banach algebraAand the conePbe the same ones as those in

Ex-ample ., and letX=R+_{. Define a mappingF:X}_{→Aas in Example .. Then (X,F)}

is anF-cone metric space over Banach algebraA. Now define the mappingT:X→Xby

T(x) =x

cos

x

. Sinceucosu≤ufor eachu∈[,∞), for allx,y∈X, we have

F(Tx,Tx,Ty)

=max{Tx,Ty}+max{Tx,Ty},αmax{Tx,Ty}+max{Tx,Ty}·et

= max{Tx,Ty},αmax{Tx,Ty}·et

=  max x cos x , y cos y   ,α max x cos x , y cos y  

·et

 max x , y   ,α max x , y  

·et

 

max{x,y},αmax{x,y}·et

= 

max{x,y}+max{x,y},αmax{x,y}+max{x,y}·et



F(x,x,y)(t),

wherek=_. Clearly,Tis a generalized Lipschitz map inX.

Now we review some facts onc-sequence theory.

Definition .([]) LetPbe a solid cone in a Banach spaceE. A sequence{un} ⊂Pis

said to be ac-sequence if for eachcθthere exists a natural numberNsuch thatun c for alln>N.

Lemma .([]) If E is a real Banach space with a solid cone P and{un} ⊂P is a sequence

Lemma .([]) Let A be a Banach algebra with a unit e,k∈A,thenlimn→∞kn



n

exists and the spectral radiusρ(k)satisfies

ρ(k) = lim

n→∞k

nn _=infknn_.

Ifρ(k) <|λ|,then(λe–k)is invertible in A;moreover,

(λe–k)–= ∞

i= ki

λi+,

whereλis a complex constant.

Lemma .([]) Let A be a Banach algebra with a unit e,a,b∈A.If a commutes with b,

then

ρ(a+b)≤ρ(a) +ρ(b),

ρ(ab)≤ρ(a)ρ(b).

Lemma .([]) If E is a real Banach space with a solid cone P

() Ifa,b,c∈Eanda≤b c,thena c. () Ifa∈Panda cfor eachcθ,thena=θ.

Lemma .([]) Let P be a solid cone in a Banach algebra A.Suppose that k∈P and

{un}is a c-sequence in P.Then{kun}is a c-sequence.

Lemma .([]) Let A be a Banach algebra with a unit e and k∈A.Ifλis a complex

constant andρ(k) <|λ|,then

ρ(λe–k)–≤ 

|λ|–ρ(k).

Lemma .([]) Let A be a Banach algebra with a unit e and P be a solid cone in A.Let

a,k,l∈P hold lk and ala.Ifρ(k) < ,then a=θ.

6 Applications to fixed point theory

In this section, we prove some famous fixed point theorems satisfying generalized Lips-chitz maps in the framework ofF-cone metric space over Banach algebraA.

Theorem . Let(X,F)be aθ-complete F-cone metric space over Banach algebra A and

suppose that T:X→X is a mapping satisfying the following condition:

F(Tx,Tx,Ty)kF(x,x,y), (.)

whereρ(k) < .Then T has a unique fixed point.For each x∈X,the sequence of iterates

{Tn_x}

Proof For eachx∈Xandn≥, setx=Txandxn+=Tn+x. Then

F(xn,xn,xn+) = F(Txn–,Txn–,Txn)

kF(xn–,xn–,xn) kF(xn–,xn–,xn–)

.. .

knF(x,x,x).

So, form>n,

F(xn,xn,xm)sF(xn,xn,xn+) +F(xn,xn,xn+) +F(xm,xm,xn+) –F(xn+,xn+,xn+)

sF(xn,xn,xn+) +F(xn+,xn+,xm)

= sF(xn,xn,xn+) +sF(xn+,xn+,xm)

sF(xn,xn,xn+) +s

F(xn+,xn+,xn+) +F(xn+,xn+,xn+)

+F(xm,xm,xn+) –F(xn+,xn+,xn+)

sF(xn,xn,xn+) + sF(xn+,xn+,xn+) +sF(xn+,xn+,xm)

sF(xn,xn,xn+) + sF(xn+,xn+,xn+) + sF(xn+,xn+,xn+)

+ – – – + sm–nF(xm–,xm–,xm)

sknF(x,x,x) + skn+F(x,x,x) + skn+F(x,x,x)

+ – – – + sm–nkm–F(x,x,x)

snkn+ sn+kn++ sn+kn++ – – – + sm–km–F(x,x,x)

= snkne+ (sk) + (sk)+ – – – + (sk)m–n–F(x,x,x)

(sk)n(e–sk)–F(x,x,x).

By Remark .,(sk)n·F(x,x,x) ≤ (sk)nF(x,x,x) →. By Lemma ., we

have{(sk)n_F(x

,x,x)}is ac-sequence. Next, by using Lemmas . and ., we conclude

that{xn}is aθ-Cauchy sequence inX.

By theθ-completeness ofX, there existsu∈Xsuch that

lim

n→∞F(xn,xn,u) =nlim→∞F(xn,xn,xm) =F(u,u,u) =θ.

Furthermore, one has

F(u,u,Tu)sF(u,u,Txn) +F(u,u,Txn) +F(Tu,Tu,Txn) –F(Txn,Txn,Txn)

sF(u,u,Txn) +F(Tu,Tu,Txn)

=sF(u,u,xn+) +F(u,u,xn)

Now that{F(u,u,xn+)}and{F(u,u,xn)}arec-sequences, by using Lemmas . and ., we

conclude thatTu=u. Thusuis a fixed point ofT.

Finally, we prove the uniqueness of the fixed point. In fact, ifvis another fixed point,

F(u,u,v) =F(Tu,Tu,Tv)

kF(u,u,v).

That is,

(e–k)F(u,u,v)θ. Multiplying both sides above by

(e–k)–= ∞

i= ki≥,

we getF(u,u,v)θ. Thus,F(u,u,v) =θ, which implies thatu=v, a contradiction.

Hence, the fixed point is unique.

Corollary . Let(X,F)be aθ-complete F-cone metric space over Banach algebra A. Sup-pose that a mapping T:X→X satisfies,for some positive integer n,

FTnx,Tnx,TnykF(x,x,y) (.)

for all x,y∈X,where k is a vector withρ(k) <

s.Then T has a unique fixed point in X.

Proof From Theorem .,Tn_{has a unique fixed pointx}∗_{. ButT}n_(Tx∗_{) =T(T}n_x∗_{) =Tx}∗_. So,Tx∗is also a fixed point ofTn. HenceTx∗=x∗,x∗is a fixed point ofT. Since the fixed point ofTis also a fixed point ofTn, then the fixed point ofTis unique.

We now prove Chatterjee’s fixed point theorem in the new space.

Theorem . Let(X,F)be aθ-complete F-cone metric space over a Banach algebra A,and let P be the underlying cone with k∈P withρ(k) <_s+ .Suppose that a mapping T:X→X satisfies the generalized Lipschitz condition

F(Tx,Tx,Ty)kF(Tx,Tx,x) +F(Ty,Ty,y) (.)

for all x,y∈X.Then T has a unique fixed point in X. And for any x∈X,the iterative sequence{Tn_{x}converges to the fixed point.}

Proof Letx∈Xbe arbitrarily given and setxn=Tnx,n≥. We have

F(xn+,xn+,xn) =F(Txn,Txn,Txn–)

kF(Txn,Txn,xn) +F(Txn–,Txn–,xn–)

kF(xn+,xn+,xn) +F(xn,xn,xn–)

which implies

(e–k)F(xn+,xn+,xn)kF(xn–,xn–,xn). (..)

Note thatρ(k) < (s+ )ρ(k) < .

Then by Lemma . it follows that (e–k) is invertible. Multiplying both sides of (..) by (e–k)–, we get

F(xn+,xn+,xn)(e–k)–·kF(xn–,xn–,xn).

As is shown in the proof of Theorem ., {xn} is aθ-Cauchy sequence, and by theθ -completeness ofX, there existsz∈Xsuch that

lim

n→∞F(xn,xn,z) =n,mlim→∞F(xn,xn,xm)

=F(z,z,z)

=θ.

Puth= (e–k)–·k,∴F(xn+,xn+,xn)hF(xn–,xn–,xn)

ρ(h) =ρ(e–k)–·k

≤ρ(e–k)–·ρ(k)

≤ ρ(k)

 –ρ(k)< 

s.

We shall show thatzis a fixed point ofT. We have

F(z,z,Tz)sF(z,z,Txn) +F(z,z,Txn) +F(Tz,Tz,Txn) –F(Txn,Txn,Txn)

sF(z,z,Txn) +F(Tz,Tz,Txn)

sF(z,z,xn+) +k

F(Tz,Tz,z) +F(Txn,Txn,xn)

= sF(z,z,xn+) +skF(z,z,Tz) +skF(xn+,xn+,xn),

which implies that

(e–sk)F(z,z,Tz)sF(z,z,xn+) +skF(xn+,xn+,xn).

Sinceρ(sk) < (s+ )ρ(k) < , it concludes by Lemma . that (e–sk) is invertible. So,

F(z,z,Tz)(e–sk)–·sF(z,z,xn+) +skF(xn+,xn+,xn)

.

Finally, we prove the uniqueness of the fixed point. In fact, ifvis another fixed point, then

F(v,v,z) =F(Tv,Tv,Tz)

kF(Tv,Tv,v) +F(Tz,Tz,z)

=kF(v,v,v) +F(z,z,z).

So,F(v,v,z)θ, which is a contradiction. HenceF(v,v,z) =θ. So,v=z.

Hence the fixed point is unique.

Example . Let the Banach algebraAand the conePbe the same ones as those in

Ex-ample ., and letX=R+_{. Define a mappingF:X}_{→Aas in Example .. We make a}

conclusion that (X,F) is aθ-completeF-cone metric space over Banach algebraA. Now define the mappingT:X→XbyT(x) =cosx_– .

Then

F(Tx,Tx,Ty)(t)

=max{Tx,Ty}+max{Tx,Ty},αmax{Tx,Ty}

+max{Tx,Ty}et

= max{Tx,Ty},αmax{Tx,Ty}et

= 

max

cosx

– ,cos

y –   ,α max cosx

– ,cos

y –   et <  max cosx

,cos

y   ,α max cosx

,cos

y   et  max x , y   ,α max x , y   et = 

max{x,y}, αmax{x,y}et

=

F(x,x,y)(t),

wherek=_, then all the conditions of Theorem . hold trivially good and  is the unique fixed point ofT. Clearly,Tis a generalized Lipschitz map inX.

7 Expansive mapping onF-cone metric space over Banach algebra

In this section, we define expansive maps inF-cone metric spaces over Banach algebra.

Definition . Let (X,F) be anF-cone metric space over Banach algebraAandPbe a

cone inA. A mapT:X→Xis said to be an expansive mapping wherek,k–_{∈Pare called}

the generalized Lipschitz constants withρ(k–_{) <  for allx,y∈Xsuch that}

Example . Let the Banach algebraAand the conePbe the same ones as those in Ex-ample ., and letX=R+_{. Define a mappingF:X}_{→Aas in Example .. Then (X,F)}

is anF-cone metric space over Banach algebraA. Now define the mappingT:X→Xby

T(x) = x. Then, for allx,y∈X, we have

F(Tx,Tx,Ty)(t)

=max{Tx,Ty}+max{Tx,Ty},αmax{Tx,Ty}+max{Tx,Ty}et

= max{Tx,Ty},αmax{Tx,Ty}et

= max{x, y},αmax{x, y}et

= max{x,y},αmax{x,y}et

max{x,y}+max{x,y},αmax{x,y}+max{x,y}et

F(x,x,y)(t),

wherek= . Clearly,Tis an expansive map inX.

Now we present a fixed point theorem for such maps.

Theorem . Let(X,F)be aθ-complete F-cone metric space over Banach algebra,and let P be an underlying solid cone with k∈P withρ((e+k+sk–a)(b+c– sk)–_{) <}

s.Let f and

g be two surjective selfmaps of X satisfying

F(fx,fx,gy) +kF(x,x,gy) +F(y,y,fx)aF(x,x,fx) +bF(y,y,gy) +cF(x,x,y) (..)

for all x,y∈X,and then f and g have a unique common fixed point in X.

Proof We define a sequencexnas follows forn= , , , , . . . :

xn=fxn+, xn+=gxn+.

Ifxn=xn+=xn+for somen, then we say thatxnis a fixed point off andg. Therefore, we suppose that no two consecutive terms of the sequence{xn}are equal.

Now, puttingx=xn+andy=xn+in (..), we get

F(fxn+,fxn+,gxn+) +k

F(xn+,xn+,gxn+) +F(xn+,xn+,fxn+)

aF(xn+,xn+,fxn+) +bF(xn+,xn+,gxn+) +cF(xn+,xn+,xn+),

F(xn,xn,xn+) +k

F(xn+,xn+,xn+) +F(xn+,xn+,xn)

aF(xn+,xn+,xn) +bF(xn+,xn+,xn+) +cF(xn+,xn+,xn+),

aF(xn+,xn+,xn) +bF(xn+,xn+,xn+) +cF(xn+,xn+,xn+)

F(xn,xn,xn+) +k

F(xn+,xn+,xn) +s

F(xn+,xn+,xn+),

+F(xn+,xn+,xn+) +F(xn,xn,xn+) –F(xn+,xn+,xn+) By (F)

,

F(xn,xn,xn+) +kF(xn+,xn+,xn) + skF(xn+,xn+,xn+)

+skF(xn,xn,xn+) –skF(xn,xn,xn+),

(b+c– sk)F(xn+,xn+,xn+)(e+k+sk–a)F(xn,xn,xn+).

Putb+c– sk=r, then

rF(xn+,xn+,xn+)(e+k+sk–a)F(xn,xn,xn+). (..)

Sinceris invertible, to multiplyr–on both sides of (..),

F(xn+,xn+,xn+)hF(xn,xn,xn+),

whereh= (e+k+sk–a)(b+c– sk)–_.

Note thatρ(h) <_s.

Hence by the proof of Theorem ., we can easily see that the sequence{xn}is aθ-Cauchy sequence. Moreover, by theθ-completeness ofX, there existsx∗∈Xsuch that

lim

n→∞F

xn,xn,x∗= lim

n,m→∞F(xn,xn,xm) =F

x∗,x∗,x∗=θ.

Sincef andgare surjective maps and hence there exist two pointsyandyinXsuch that

x∗=fyandx∗=gy. Consider

Fxn,xn,x∗

=Ffxn+,fxn+,gy

–kFxn+,xn+,gy

+Fy,y,fxn+

+aF(xn+,xn+,fxn+)

+bFy,y,gy+cFxn+,xn+,y

.

Then

Fxn,xn,x∗

–kFxn+,xn+,x∗

–kFy,y,xn

+aF(xn+,xn+,xn)

+bFy,y,x∗+cFxn+,xn+,y

.

Since

Fy,y,x∗sFy,y,xn+

+Fx∗,x∗,xn+

–F(xn+,xn+,xn+)

,

so

–kFxn+,xn+,x∗

–kFy,y,xn

+aF(xn+,xn+,xn) +cF

xn+,xn+,y

Fxn,xn,x∗

–bsFy,y,xn+

+sFx∗,x∗,xn+

–sF(xn+,xn+,xn+)

,

–kFxn+,xn+,x∗

+aF(xn+,xn+,xn) +cF

xn+,xn+,y

ksF(xn,xn,xn+) +F

xn+,xn+,y

–F(xn+,xn+,xn+)

+sF(xn,xn,xn+) +F

xn+,xn+,x∗

–bsFy,y,xn+

+sFx∗,x∗,xn+

–F(xn+,xn+,xn+)

,

which implies

(sb+c–sk)Fxn+,xn+,y

(s–bs+k)Fxn+,xn+,x∗

+ (sk+ s–a)F(xn+,xn+,xn).

Since sb+c–sk=ris invertible, we have

rFxn+,xn+,y

(s–bs+k)·Fxn+,xn+,x∗

+ (sk+ s–a)·F(xn+,xn+,xn),

Fxn+,xn+,y

r–(s–bs+k)Fxn+,xn+,x∗

+ (sk+ s–a)F(xn+,xn+,xn)

.

Now that{F(xn+,xn+,x∗)}and{F(xn+,xn+,xn)}arec-sequences, then by using Lem-mas . and ., we conclude thatgxn+=y.

Finally, we prove the uniqueness of the fixed point. In fact, ify∗is another common fixed point off andg, that is,fy∗=y∗andgy∗=y∗,

Fx,x,y∗

=FTx,Tx,Ty∗

–kFx,x,gy∗+Fy∗,y∗,fx+aF(x,x,fx) +bFy∗,y∗,gy∗+cFx,x,y∗

⇒ Fx,x,y∗ –kFx,x,y∗+Fy∗,y∗,x+aF(x,x,x) +bFy∗,y∗,y∗

+cFx,x,y∗

⇒ Fx,x,y∗(–k+c)Fx,x,y∗

or

(c– k)Fx,x,y∗Fx,x,y∗,

(c– k–e)Fx,x,y∗θ,

which means F(x,x,y∗) =θ, which implies thatx=y∗, a contradiction. Hence the fixed

point is unique.

Corollary . Let(X,F)be aθ-complete F-cone metric space over Banach algebra,and

let P be an underlying solid cone,where c∈P is a generalized Lipschitz constant with

ρ(c)–_<

s.Let f and g be two surjective selfmaps of X satisfying

F(fx,fx,gy)cF(x,x,y). (.)

Proof If we putk,a,b=θin Theorem ., then we get the above Corollary ..

Corollary . Let(X,F)be aθ-complete F-cone metric space over Banach algebra,and

let P be an underlying solid cone,where c∈P is a generalized Lipschitz constant with

ρ(c)–<_s.Let f be a surjective selfmap of X satisfying

F(fx,fx,fy)cF(x,x,y). (.)

Then f has a unique fixed point in X.

Proof If we putf =g in Corollary ., then we get the above Corollary . which is an extension of Theorem  of Wang et al. [] in anF-cone metric space over Banach

alge-bra.

Corollary . Let(X,F)be aθ-complete F-cone metric space over Banach algebra,and

let P be an underlying solid cone,where c∈P is a generalized Lipschitz constant with

ρ(c)–<_s.Let f be a surjective selfmap of X,and suppose that there exists a positive in-teger n satisfying

Ffnx,fnx,fnycF(x,x,y). (.)

Then f has a unique fixed point in X.

Proof From Corollary .fnhas a unique fixed pointz. Butfn(fz) =f(fnz) =fz, sofzis also a fixed point offn_{. Hencefz=z,zis a fixed point off. Since the fixed point off is also a} fixed point offn_{, the fixed point off is unique.}

Corollary . Let(X,F)be aθ-complete F-cone metric space over Banach algebra,and

let P be an underlying solid cone,where a,b,c, –a∈P are generalized Lipschitz constants withρ[(e–a)(b+c)–] <_s.Let f and g be two surjective selfmaps of X satisfying

F(fx,fx,gy)aF(x,x,fx) +bF(y,y,gy) +cF(x,x,y). (.)

Then f and g have a unique common fixed point in X.

Proof If we putk=θin Theorem ., then we get the above Corollary ..

Corollary . Let(X,F)be aθ-complete F-cone metric space over Banach algebra,and

let P be an underlying solid cone,where a,b,c, –a∈P are generalized Lipschitz constants withρ[(e–a)(b+c)–_{] <}

s.Let f be a surjective selfmap of X satisfying

F(fx,fx,fy)aF(x,x,fx) +bF(y,y,fy) +cF(x,x,y). (.)

Then f has a unique fixed point in X.

Proof If we putf =g in Corollary ., then we get the above Corollary . which is an extension of Theorem  of Wang et al. [] in anF-cone metric space over Banach

8 Conclusion

In this paper, we introduce anF-cone metric space over Banach algebra which generalizes anNp-cone metric space over Banach algebra and anNb-cone metric space over Banach algebra. We introduce the concept of generalized Lipschitz and expansive mapping in the new structure. Also we derive the existence and uniqueness of some fixed point theorems for such spaces. Our main theorems extend and unify the existing results in the recent literature. Example is constructed to support our result.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, NRI Institute of Information Science and Technology, Bhopal, (M.P.), 462021, India.} 2_{Nonlinear Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam.}3_{Faculty of Mathematics and}

Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam. 4_{Department of Mathematics, Government M V M,}

Bhopal, (M.P.), 462001, India.

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Received: 23 December 2016 Accepted: 28 April 2017 References

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