# Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality

- Shaoyuan Xu
^{1}Email author and - Stojan Radenović
^{2}

**2014**:102

https://doi.org/10.1186/1687-1812-2014-102

© Xu and Radenović; licensee Springer. 2014

**Received: **27 December 2013

**Accepted: **17 April 2014

**Published: **6 May 2014

## Abstract

In this paper, we first present some elementary results concerning cone metric spaces over Banach algebras. Next, by using these results and the related ones about *c*-sequence on cone metric spaces we obtain some new fixed point theorems for the generalized Lipschitz mappings on cone metric spaces over Banach algebras without the assumption of normality. As a consequence, our main results improve and generalize the corresponding results in the recent paper by Liu and Xu (Fixed Point Theory Appl. 2013:320, 2013).

**MSC:**54H25, 47H10.

## Keywords

*c*-sequencesspectral radius

## 1 Introduction

In 2007, cone metric spaces were reviewed by Huang and Zhang, as a generalization of metric spaces (see [1]). The distance $d(x,y)$ of two elements *x* and *y* in a cone metric space *X* is defined to be a vector in an ordered Banach space *E*, quite different from that which is defined to be a non-negative real number in general metric spaces. They gave the version of the Banach contraction principle and other basic theorems in the setting of cone metric spaces. Later on, by omitting the assumption of normality in the results of [1], Rezapour and Hamlbarani [2] obtained some fixed point theorems, as the generalizations of the relevant results in [1]. Besides, they presented a number of examples to prove the existence of non-normal cones, which shows that such generalizations are meaningful. Since then, many authors have been interested in the study of fixed point results in the setting of cone metric spaces (see [2–10]).

The right-hand side of the inequality (1) is the vector as the result of the operation of scalar multiplication in cone metric spaces. In [1], the authors proved that there exists a unique fixed point for contractive mappings in complete cone metric spaces.

Recently, some authors investigated the problem of whether cone metric spaces are equivalent to metric spaces in terms of the existence of the fixed points of the mappings involved. They established the equivalence between some fixed point results in metric and in (topological vector space-valued) cone metric spaces see [11–14]. Actually, they showed that any cone metric space $(X,d)$ is equivalent to a usual metric space $(X,{d}^{\ast})$, where the real-valued metric function ${d}^{\ast}$ is defined by a nonlinear scalarization function ${\xi}_{e}$ (see [12]) or by a Minkowski functional ${q}_{e}$ (see [13]). After that, some other interesting generalizations were developed. See, for instance, [4].

Very recently, Liu and Xu [15] introduced the concept of cone metric spaces over Banach algebras (which were called cone metric spaces over Banach algebras in [15]), replacing Banach spaces by Banach algebras as the underlying spaces of cone metric spaces. They replaced the Banach space *E* by a Banach algebra
and introduced the concept of cone metric spaces over Banach algebras. In this way, they proved some fixed point theorems of generalized Lipschitz mappings with weaker and natural conditions on generalized Lipschitz constant *k* by means of spectral radius. Note that it is significant to introduce the concept of cone metric spaces with Banach algebras since one can prove that cone metric spaces with Banach algebras are not equivalent to metric spaces in terms of the existence of the fixed points of the generalized Lipschitz mappings. As a matter of fact, Liu and Xu showed that the main results obtained in [15] could not be reduced to a consequence of corresponding results in metric spaces by means of the methods in the literature. This does bring about prosperity in the study of cone metric spaces. However, the proofs of the main results in [15] depends strongly on the condition that the underlying solid cone is normal. In this paper, we delete the superfluous assumption of normality of the paper [15] and also obtain the existence and uniqueness of the fixed point for the generalized Lipschitz mappings in the setting of cone metric spaces over Banach algebras. The methods and techniques used in this paper are quite different from those appearing in [2]. Furthermore, we give an example to support that it is meaningful to set up fixed point theorems of generalized Lipschitz mappings without the assumption of normality of the underlying solid cones.

## 2 Preliminaries

- 1.
$(xy)z=x(yz)$;

- 2.
$x(y+z)=xy+xz$ and $(x+y)z=xz+yz$;

- 3.
$\alpha (xy)=(\alpha x)y=x(\alpha y)$;

- 4.
$\parallel xy\parallel \u2a7d\parallel x\parallel \parallel y\parallel $.

Throughout this paper, we shall assume that a Banach algebra has a unit (*i.e.*, a multiplicative identity) *e* such that $ex=xe=x$ for all $x\in \mathcal{A}$. An element $x\in \mathcal{A}$ is said to be invertible if there is an inverse element $y\in \mathcal{A}$ such that $xy=yx=e$. The inverse of *x* is denoted by ${x}^{-1}$. For more details, we refer to [16].

The following proposition is well known (see [16]).

**Proposition 2.1**

*Let*

*be a Banach algebra with a unit*

*e*,

*and*$x\in \mathcal{A}$.

*If the spectral radius*$r(x)$

*of*

*x*

*is less than*1,

*i*.

*e*.,

*then*$e-x$

*is invertible*.

*Actually*,

**Remark 2.1**From [16] we see that the spectral radius $r(x)$ of

*x*satisfies

for all $x\in \mathcal{A}$, where
is a Banach algebra with a unit *e*.

**Remark 2.2** In Proposition 2.1, if the condition ‘$r(x)<1$’ is replaced by $\parallel x\parallel \le 1$, then the conclusion remains true.

*P*of is called a cone of if

- 1.
*P*is non-empty closed and $\{\theta ,e\}\subset P$; - 2.
$\alpha P+\beta P\subset P$ for all non-negative real numbers

*α*,*β*; - 3.
${P}^{2}=PP\subset P$;

- 4.
$P\cap (-P)=\{\theta \}$,

where *θ* denotes the null of the Banach algebra
. For a given cone $P\subset \mathcal{A}$, we can define a partial ordering ⪯ with respect to *P* by $x\u2aafy$ if and only if $y-x\in P$. $x\prec y$ will stand for $x\u2aafy$ and $x\ne y$, while $x\ll y$ will stand for $y-x\in intP$, where int*P* denotes the interior of *P*. If $intP\ne \mathrm{\varnothing}$ then *P* is called a solid cone.

*P*is called normal if there is a number $M>0$ such that, for all $x,y\in \mathcal{A}$,

The least positive number satisfying above is called the normal constant of *P* [1].

In the following we always assume that *P* is a cone in
with $intP\ne \mathrm{\varnothing}$ and ⪯ is the partial ordering with respect to *P*.

**Definition 2.1** (See [1, 15] and [17])

*X*be a non-empty set. Suppose that the mapping $d:X\times X\to \mathcal{A}$ satisfies

- 1.
$0\u2aafd(x,y)$ for all $x,y\in X$ and $d(x,y)=0$ if and only if $x=y$;

- 2.
$d(x,y)=d(y,x)$ for all $x,y\in X$;

- 3.
$d(x,y)\u2aafd(x,z)+d(z,x)$ for all $x,y,z\in X$.

Then *d* is called a cone metric on *X*, and $(X,d)$ is called a cone metric space over a Banach algebra
.

For convenance, we give an example of cone metric space over Banach algebra as follows. For more examples, see [15].

**Example 2.1** (See [15])

*n*-square real matrices, and define the norm

Then
is a real Banach algebra with the unit *e* the identity matrix.

Let $P=\{a\in \mathcal{A}\mid {a}_{ij}\u2a7e0,\text{for all}1\u2a7di,j\u2a7dn\}$. Then $P\subset \mathcal{A}$ is a normal cone with normal constant $M=1$.

Then $(X,d)$ is a cone metric space over Banach algebra with normality.

**Example 2.2** Let $\mathcal{A}={C}_{\mathbb{R}}^{1}[0,1]$ and define a norm on
by $\parallel x\parallel ={\parallel x\parallel}_{\mathrm{\infty}}+{\parallel {x}^{\prime}\parallel}_{\mathrm{\infty}}$ for $x\in \mathcal{A}$. Define multiplication in
as just pointwise multiplication. Then
is a real unital Banach algebra with unit $e=1$. The set $P=\{x\in \mathcal{A}:x\ge 0\}$ is a cone in
. Moreover, *P* is not normal (see [2]).

Let $X=\{1,2,3\}$. Define $d:X\times X\to \mathcal{A}$ by $d(1,2)(t)=d(2,1)(t)=d(2,3)(t)=d(3,2)(t)={e}^{t}$, $d(1,3)(t)=d(3,1)(t)=2{e}^{t}$, $d(x,x)(t)=0$. We see that $(X,d)$ is a cone metric space over Banach algebra without normality.

**Definition 2.2** (See [1, 15] or [17])

*X*. Then:

- 1.
$\{{x}_{n}\}$ converges to

*x*whenever for each $c\in \mathcal{A}$ with $\theta \ll c$ there is a natural number*N*such that $d({x}_{n},x)\ll c$ for all $n\u2a7eN$. We denote this by ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$ or ${x}_{n}\to x$. - 2.
$\{{x}_{n}\}$ is a Cauchy sequence whenever for each $c\in \mathcal{A}$ with $\theta \ll c$ there is a natural number

*N*such that $d({x}_{n},{x}_{m})\ll c$ for all $n,m\u2a7eN$. - 3.
$(X,d)$ is a complete cone metric space if every Cauchy sequence is convergent.

Now, we shall appeal to the following lemmas in the sequel.

**Lemma 2.1** (See [18])

*If* *E* *is a real Banach space with a cone* *P* *and if* $a\u2aaf\lambda a$ *with* $a\in P$ *and* $0\le \lambda <1$, *then* $a=\theta $.

**Lemma 2.2** (See [9])

*If* *E* *is a real Banach space with a solid cone* *P* *and if* $\theta \u2aafu\ll c$ *for each* $\theta \ll c$, *then* $u=\theta $.

**Lemma 2.3** (See [9])

*If* *E* *is a real Banach space with a solid cone* *P* *and if* $\parallel {x}_{n}\parallel \to 0$ ($n\to \mathrm{\infty}$), *then for any* $\theta \ll c$, *there exists* $N\in \mathbb{N}$ *such that*, *for any* $n>N$, *we have* ${x}_{n}\ll c$.

Finally, let us recall the concept of generalized Lipschitz mapping defining on the cone metric spaces over Banach algebras, which is introduced in [15].

**Definition 2.3** (See [15])

**Remark 2.3** In Definition 2.3, we only suppose the spectral radius of *k* is less than 1, while $\parallel k\parallel <1$ is not assumed. Generally speaking, it is meaningful since by Remark 2.1, the condition $r(k)<1$ is weaker than that $\parallel k\parallel <1$.

**Remark 2.4** If $r(k)<1$ then $\parallel {k}^{n}\parallel \to 0$ ($n\to \mathrm{\infty}$).

## 3 Main results

In this section, by omitting the assumption of normality of the main results in Liu and Xu [15], we shall prove some fixed point theorems of generalized Lipschitz mappings in the setting of cone metric spaces over Banach algebras.

We begin this section with reviewing some facts on *c*-sequence theory.

Let *P* be a solid cone in a Banach space
. A sequence $\{{u}_{n}\}\subset P$ is a *c*-sequence if for each $c\gg \theta $ there exists ${n}_{0}\in \mathbb{N}$ such that ${u}_{n}\ll c$ for $n\ge {n}_{0}$.

It is easy to show the following proposition.

**Proposition 3.1** (See [19])

*Let* *P* *be a solid cone in a Banach space*
*and let* $\{{x}_{n}\}$ *and* $\{{y}_{n}\}$ *be sequences in* *P*. *If* $\{{x}_{n}\}$ *and* $\{{y}_{n}\}$ *are* *c*-*sequences and* $\alpha ,\beta >0$, *then* $\{\alpha {x}_{n}+\beta {y}_{n}\}$ *is a* *c*-*sequence*.

In addition to Proposition 3.1 above, the following propositions are crucial to the proof of our main result.

**Proposition 3.2** (See [19])

*Let*

*P*

*be a solid cone in a Banach algebra*

*and let*$\{{x}_{n}\}$

*be a sequence in*

*P*.

*Then the following conditions are equivalent*:

- (1)
$\{{x}_{n}\}$

*is a**c*-*sequence*. - (2)
*For each*$c\gg \theta $*there exists*${n}_{0}\in \mathbb{N}$*such that*${x}_{n}\prec c$*for*$n\ge {n}_{0}$. - (3)
*For each*$c\gg \theta $*there exists*${n}_{1}\in \mathbb{N}$*such that*${x}_{n}\u2aafc$*for*$n\ge {n}_{1}$.

**Proposition 3.3** *Let* *P* *be a solid cone in a Banach algebra*
*and let* $\{{u}_{n}\}$ *be a sequence in* *P*. *Suppose that* $k\in P$ *is an arbitrarily given vector and* $\{{u}_{n}\}$ *is a* *c*-*sequence in* *P*. *Then* $\{k{u}_{n}\}$ *is a* *c*-*sequence*.

*Proof* Fix $c\gg \theta $. Then also $\frac{c}{m}\gg \theta $ for all $m\in \mathbb{N}$. Therefore, since $\{{x}_{n}\}$ is a *c*-sequence, it follows that, for each $m\in \mathbb{N}$, there exists ${n}_{m}\in \mathbb{N}$ such that ${u}_{n}\ll \frac{c}{m}$ for all $n\ge {n}_{m}$. Hence, for all $n\ge {n}_{m}$ we have $k{u}_{n}\u2aaf\frac{kc}{m}$. Now, since $\frac{kc}{m}\to \theta $ as $m\to \mathrm{\infty}$, there exists ${m}_{1}\in \mathbb{N}$ such that $\frac{kc}{m}\ll c$ for all $m>{m}_{1}$. Hence, there exists ${n}_{0}\in \mathbb{N}$ such that, for all $n>{n}_{0}$ we have $k{u}_{n}\ll c$, that is, $\{k{u}_{n}\}$ is a *c*-sequence. □

**Proposition 3.4**

*Let*

*be a Banach algebra with a unit*

*e*,

*P*

*be a cone in*

*and*⪯

*be the semi*-

*order generated by the cone*

*P*.

*The following assertions hold true*:

- (i)
*For any*$x,y\in \mathcal{A}$, $a\in P$*with*$x\u2aafy$,*we have*$ax\u2aafay$. - (ii)
*For any sequences*$\{{x}_{n}\},\{{y}_{n}\}\subset \mathcal{A}$*with*${x}_{n}\to x$ ($n\to \mathrm{\infty}$)*and*${y}_{n}\to y$ ($n\to \mathrm{\infty}$)*where*$x,y\in \mathcal{A}$,*we have*${x}_{n}{y}_{n}\to xy$ ($n\to \mathrm{\infty}$).

**Proposition 3.5**

*Let*

*be a Banach algebra with a unit*

*e*,

*P*

*be a cone in*

*and*⪯

*be the semi*-

*order generated by the cone*

*P*.

*Let*$\lambda \in P$.

*If the spectral radius*$r(\lambda )$

*of*

*λ*

*is less than*1,

*then the following assertions hold true*:

- (i)
*Suppose that**x**is invertible and that*${x}^{-1}\succ \theta $*implies*$x\succ \theta $,*then for any integer*$n\ge 1$,*we have*${\lambda}^{n}\u2aaf\lambda \u2aafe$. - (ii)
*For any*$u>\theta $,*we have*$u\u22e0\lambda u$,*i*.*e*., $\lambda u-u\notin P$. - (iii)
*If*$\lambda \u2ab0\theta $,*then we have*${(e-\lambda )}^{-1}\u2ab0\theta $.

*Proof*(i) Since $r(\lambda )<1$, by Proposition 2.1, we see the element $e-\lambda $ is invertible. Considering

for all $n\ge 1$ by induction. Therefore, the conclusion of (i) is true.

- (iii)
It is obvious. □

**Remark 3.1** Proposition 3.5(ii) is the version of Lemma 2.1 in the setting of cone metric spaces over Banach algebras.

It is easy to show the following proposition by Definitions 2.1 and 3.1, so we omit its proof.

**Proposition 3.6**

*Let*$(X,d)$

*be a complete cone metric space over a Banach algebra*

*and let*

*P*

*be the underlying solid cone in Banach algebra*.

*Let*$\{{x}_{n}\}$

*be a sequence in*

*X*.

*If*$\{{x}_{n}\}$

*converges to*$x\in X$,

*then we have*:

- (i)
$\{d({x}_{n},x)\}$

*is a**c*-*sequence*. - (ii)
*For any*$p\in \mathbb{N}$, $\{d({x}_{n},{x}_{n+p})\}$*is a**c*-*sequence*.

Now, we begin to present the version of the famous Banach contraction principle for generalized Lipschitz mappings in the setting of cone metric space over Banach algebra without the assumption of normality of the underlying solid cone.

**Theorem 3.1**

*Let*$(X,d)$

*be a complete cone metric space over a Banach algebra*

*and let*

*P*

*be the underlying solid cone with*$k\in P$

*where*$r(k)<1$.

*Suppose the mapping*$T:X\to X$

*satisfies generalized Lipschitz condition*:

*Then* *T* *has a unique fixed point in* *X*. *For any* $x\in X$, *iterative sequence* $\{{T}^{n}x\}$ *converges to the fixed point*.

*Proof*Let $x\in X$ be arbitrarily given and set ${x}_{n}={T}^{n}x$, $n\u2a7e1$. We have

which implies that $\{{x}_{n}\}$ is a Cauchy sequence.

*X*, there exists ${x}^{\ast}\in X$ such that ${x}_{n}\to {x}^{\ast}$ ($n\to \mathrm{\infty}$). Furthermore, one has

Therefore, it follows from Definition 3.1, Propositions 3.1, 3.2, 3.3, and 3.6 that we have $d({x}^{\ast},T{x}^{\ast})\u2aaf{y}_{n}$ where ${y}_{n}$ is a *c*-sequence in cone *P*. Hence, for each $c\gg \theta $ we have $d({x}^{\ast},T{x}^{\ast})\ll c$, so $d({x}^{\ast},T{x}^{\ast})=\theta $ by Lemma 2.2. Thus ${x}^{\ast}$ is a fixed point of *T*.

Finally, we prove the uniqueness of the fixed point. The proof is as same as that in Theorem 2.1 in [15] since it actually does not require the assumption of normality of the underlying solid cone. □

In the following, we will present some other fixed point theorems of generalized Lipschitz mappings in the setting of cone metric space over Banach algebra without the assumption of normality of the underlying solid cone. For convenience, let us give some basic results concerning spectral radius.

**Lemma 3.1**

*Let*

*be a Banach algebra and let*

*x*,

*y*

*be vectors in*.

*If*

*x*

*and*

*y*

*commute*,

*then the following hold*:

- (i)
$r(xy)\le r(x)r(y)$;

- (ii)
$r(x+y)\le r(x)+r(y)$;

- (iii)
$|r(x)-r(y)|\le r(x-y)$.

**Lemma 3.2** *Let*
*be a Banach algebra and let* $\{{x}_{n}\}$ *be a sequence in*
. *Suppose that* $\{{x}_{n}\}$ *converges to* *x* *in*
*and that* ${x}_{n}$ *and* *x* *commute for all* *n*, *then we have* $r({x}_{n})\to r(x)$ *as* $n\to \mathrm{\infty}$.

*Proof*It follows from Lemma 3.1 and Remark 2.1 that

which completes the proof. □

**Lemma 3.3** *Let*
*be a Banach algebra and let* *k* *be a vector in*
. *If* $0\le r(k)<1$, *then we have* $r({(e-k)}^{-1})\le {(1-r(k))}^{-1}$.

*Proof*Since $0\le r(k)<1$, it follows that

which completes the proof. □

**Theorem 3.2**

*Let*$(X,d)$

*be a complete cone metric space over a Banach algebra*

*and let*

*P*

*be the underlying solid cone with*$k\in P$

*where*$r(k)<\frac{1}{2}$.

*Suppose the mapping*$T:X\to X$

*satisfies the generalized Lipschitz condition*

*Then* *T* *has a unique fixed point in* *X*. *And for any* $x\in X$, *iterative sequence* $\{{T}^{n}x\}$ *converges to the fixed point*.

*Proof*Let $x\in X$ be arbitrarily given and set ${x}_{n}={T}^{n}x$, $n\u2a7e1$. We have

*k*commute, by Lemma 3.3 above and the fact that $r(k)<\frac{1}{2}$, we have

Hence, by the proof of Theorem 3.1, we can easily see that the sequence $\{{x}_{n}\}$ is Cauchy.

*X*, there is ${x}^{\ast}\in X$ such that ${x}_{n}\to {x}^{\ast}$ ($n\to \mathrm{\infty}$). To verify $T{x}^{\ast}={x}^{\ast}$, we have

Therefore, it follows from Definition 3.1, Propositions 3.1, 3.2, 3.3, and 3.6 that we have $d({x}^{\ast},T{x}^{\ast})\u2aaf{y}_{n}$ where ${y}_{n}$ is a *c*-sequence in cone *P*. Hence, for each $c\gg \theta $ we have $d({x}^{\ast},T{x}^{\ast})\ll c$, so $d({x}^{\ast},T{x}^{\ast})=\theta $ by Lemma 2.2. Thus ${x}^{\ast}$ is a fixed point of *T*.

for any $n\u2a7e1$.

which implies by Lemma 2.2 that $d({x}^{\ast},{y}^{\ast})=0$, so ${x}^{\ast}={y}^{\ast}$, a contradiction. Hence, the fixed point is unique. □

**Theorem 3.3**

*Let*$(X,d)$

*be a complete cone metric space over a Banach algebra*

*and let*

*P*

*be the underlying solid cone with*$k\in P$

*where*$r(k)<\frac{1}{2}$.

*Suppose the mapping*$T:X\to X$

*satisfies the generalized Lipschitz condition*

*Then* *T* *has a unique fixed point in* *X*. *And for any* $x\in X$, *iterative sequence* $\{{T}^{n}x\}$ *converges to the fixed point*.

*Proof*Let $x\in X$ be arbitrarily given and set ${x}_{n}={T}^{n}x$, $n\u2a7e1$. We have

As is shown in the proof of Theorem 3.2, $\{{x}_{n}\}$ is a Cauchy sequence, and, by the completeness of *X*, the limit of $\{{x}_{n}\}$ exists and is denoted by ${x}^{\ast}$.

*T*, we have

Therefore, it follows from Definition 3.1, Propositions 3.1, 3.2, 3.3, and 3.6 that we have $d({x}^{\ast},T{x}^{\ast})\u2aaf{y}_{n}$ where ${y}_{n}$ is a *c*-sequence in cone *P*. Hence, for each $c\gg \theta $ we have $d({x}^{\ast},T{x}^{\ast})\ll c$, so $d({x}^{\ast},T{x}^{\ast})=\theta $ by Lemma 2.2. Thus ${x}^{\ast}$ is a fixed point of *T*.

Finally, we prove the uniqueness of the fixed point. The proof is as same as that in Theorem 2.3 in [15] since it actually does not require the assumption of normality of the underlying solid cone. □

**Remark 3.2**The method and technique to prove the inequality

in Theorems 3.2 and 3.3 in this paper are more simple than and different from those in Theorems 2.2 and 2.3 in [15].

We conclude the paper with two examples.

**Example 3.1** Let $\mathcal{A}={C}_{\mathbb{R}}^{1}[0,1]$ be the same as that in Example 2.2. Then the set $P=\{x\in \mathcal{A}:x\ge 0\}$ is a non-normal cone in
.

Let $X=\{1,2,3\}$. Define $d:X\times X\to \mathcal{A}$ by $d(1,2)(t)=d(2,1)(t)={e}^{t}$, $d(2,3)(t)=d(3,2)(t)=2{e}^{t}$, $d(3,1)(t)=d(1,3)(t)=3{e}^{t}$, for all $t\in [0,1]$ and $d(x,x)(t)=\theta $ for each $x\in X$ and for all $t\in [0,1]$. We find that $(X,d)$ is a solid cone metric space over Banach algebra
. Further, let $T:X\to X$ be a mapping defined with $T1=T2=2$, $T3=1$ and let $k\in P$ defined with $k(t)=\frac{1}{3}t+\frac{1}{2}$. By careful calculations one sees that all the conditions of Theorem 3.1 are fulfilled. The point $x=2$ is the unique fixed point of the map *T*.

**Example 3.2** [15]

Then is a Banach algebra with unit $e=(1,0)$.

Let $P=\{({x}_{1},{x}_{2})\in {\mathbb{R}}^{2}\mid {x}_{1},{x}_{2}\u2a7e0\}$. Then *P* is normal with normal constant $M=1$.

*d*be defined by

Then $(X,d)$ is a complete cone metric space over a Banach algebra .

*α*can be any large positive real number. Then we have

where the spectral radius of $(\frac{1}{2},\alpha )$ satisfies $r((\frac{1}{2},\alpha ))<1$. In addition, all the other conditions of Theorem 3.1 are fulfilled. By Theorem 3.1, *T* has a unique fixed point in *X*.

**Remark 3.3** Since in Example 3.1 the underlying solid cone *P* in the Banach algebra
is not normal, we can conclude that any of the theorems in [15] cannot cope with Example 3.1, which shows that the main results without the assumption of normality in this paper are meaningful.

**Remark 3.4** In Example 3.2, we see that the main results in this paper are indeed more different than the standard results of cone metric spaces presented in the literature.

## Declarations

### Acknowledgements

The authors are grateful to the referees and the editors for valuable comments and suggestions, which have improved the original manuscript greatly. The research is partially supported by Doctoral Initial Foundation of Hanshan Normal University, China (No. QD20110920).

## Authors’ Affiliations

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