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# Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings

- Hao Liu
^{1}and - Shaoyuan Xu
^{2}Email author

**2013**:320

https://doi.org/10.1186/1687-1812-2013-320

© Liu and Xu; licensee Springer. 2013

**Received:**28 July 2013**Accepted:**29 October 2013**Published:**25 November 2013

## Abstract

In this paper we introduce the concept of cone metric spaces with Banach algebras, replacing Banach spaces by Banach algebras as the underlying spaces of cone metric spaces. With this modification, we shall prove some fixed point theorems of generalized Lipschitz mappings with weaker conditions on generalized Lipschitz constants. An example shows that our main results concerning the fixed point theorems in the setting of cone metric spaces with Banach algebras are more useful than the standard results in cone metric spaces presented in the literature.

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

## Keywords

- cone metric spaces with Banach algebras
- fixed point theorems
- generalized Lipschitz conditions

## 1 Introduction

*x*and

*y*in a cone metric space

*X*is defined to be a vector in an ordered Banach space

*E*, and a mapping $T:X\to X$ is said to be contractive if there is a constant $k\in [0,1)$ such that

*T*in $(X,d)$ satisfying (1), one can get

which implies that cone metric spaces are a special case of classical metric spaces. After that, some other interesting generalizations were developed. See, for instance, [5].

In this paper, we replace the Banach space *E* by a Banach algebra *A* and obtain the concept of cone metric spaces with Banach algebras. In this way, we shall prove some fixed point theorems of generalized Lipschitz mappings with weaker and natural conditions on the Lipschitz constant *k*. Our results generalize metric spaces and reveal the fact that the essential conditions on the contraction constant *k* are neither order relations nor norm relations, but spectrum radius.

*A*always be a real Banach algebra. That is,

*A*is a real Banach space in which an operation of multiplication is defined, subject to the following properties (for all $x,y,z\in A$, $\alpha \in \mathbb{R}$):

- 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 $.

In 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 $x\in A$. An element $x\in A$ is said to be invertible if there is an inverse element $y\in A$ such that $xy=yx=e$. The inverse of *x* is denoted by ${x}^{-1}$. For more details, we refer to [6].

The following proposition is well known (see [6]).

**Proposition 1.1**

*Let*

*A*

*be a Banach algebra with a unit*

*e*,

*and*$x\in A$.

*If the spectral radius*$\rho (x)$

*of*

*x*

*is less than*1,

*i*.

*e*.,

*then*$e-x$

*is invertible*.

*Actually*,

*P*of

*A*is called a cone if

- 1.
*P*is non-empty closed and $\{0,e\}\subset P$; - 2.
$\alpha P+\beta P\subset P$ for all non-negative real numbers

*α*,*β*; - 3.
${P}^{2}=PP\subset P$;

- 4.
$P\cap (-P)=\{0\}$.

For a given cone $P\subset A$, we can define a partial ordering ⩽ with respect to *P* by $x\u2a7dy$ if and only if $y-x\in P$. $x\lneqq y$ will stand for $x\u2a7dy$ and $x\ne y$. While $x<y$ will stand for $y-x\in intP$, where int*P* denotes the interior of P.

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

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

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

**Definition 1.1** (See [1])

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

- 1.
$0\u2a7dd(x,y)$ 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.
$d(x,y)\u2a7dd(x,z)+d(z,x)$ 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 (with a Banach algebra *A*).

We present some examples in the following.

**Example 1.1**Let $A={M}_{n}(\mathbb{R})=\{a={({a}_{ij})}_{n\times n}\mid {a}_{ij}\in \mathbb{R}\text{for all}1\u2a7di,j\u2a7dn\}$ be the algebra of all

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

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

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

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

**Example 1.2** Let *A* be the Banach space $C(K)$ of all continuous real-valued functions on a compact Hausdorff topological space *K*, with multiplication defined pointwise. Then *A* is a Banach algebra, and the constant function $f(t)=1$ is the unit of *A*.

Let $P=\{f\in A\mid f(t)\u2a7e0\text{for all}t\in K\}$. Then $P\subset A$ is a normal cone with a normal constant $M=1$.

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

**Example 1.3**Let $A={\ell}^{1}=\{a={({a}_{n})}_{n\u2a7e0}\mid {\sum}_{n=0}^{\mathrm{\infty}}|{a}_{n}|<\mathrm{\infty}\}$ with convolution as multiplication:

Thus *A* is a Banach algebra. The unit *e* is $(1,0,0,\dots )$.

Let $P=\{a={({a}_{n})}_{n\u2a7e0}\in A\mid {a}_{n}\u2a7e0\text{for all}n\u2a7e0\}$, which is a normal cone in *A*.

Then $(X,d)$ is a cone metric space with *A*.

**Definition 1.2** (See [1])

*X*. Then

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

*x*whenever for each $c\in A$ with $0<c$, there is a natural number*N*such that $d({x}_{n},x)<c$ 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 $c\in A$ with $0<c$, there is a natural number

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

Finally, we shall appeal to the following lemmas in the sequel [1].

**Lemma 1.1** *Let* $(X,d)$ *be a cone metric space*, *P* *be a normal cone with a normal constant* *M*. *Let* $\{{x}_{n}\}$ *be a sequence in* *X*. *Then* $\{{x}_{n}\}$ *converges to* *x* *if and only if* $d({x}_{n},x)\to 0$ ($n\to \mathrm{\infty}$).

**Lemma 1.2** *Let* $(X,d)$ *be a cone metric space*, *P* *be a normal cone with a normal constant* *M*. *Let* $\{{x}_{n}\}$ *be a sequence in* *X*. *Then* $\{{x}_{n}\}$ *is a Cauchy sequence if and only if* $d({x}_{n},{x}_{m})\to 0$ ($n,m\to \mathrm{\infty}$).

## 2 Main results

In this section we shall prove some fixed point theorems of generalized Lipschitz mappings in the setting of cone metric spaces with Banach algebras.

**Theorem 2.1**

*Let*$(X,d)$

*be a complete cone metric space and*

*P*

*be a normal cone with a normal constant*

*M*.

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

*satisfies the generalized Lipschitz condition*

*where* $k\in P$ *with* $\rho (k)<1$. *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*Choose ${x}_{0}\in X$ and set ${x}_{n}={T}^{n}x$, $n\u2a7e1$. We have

*P*is normal with a normal constant

*M*, and note that $\parallel {k}^{n}\parallel \to 0$ ($n\to \mathrm{\infty}$), we have

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

Hence $\parallel d(T{x}^{\ast},{x}^{\ast})\parallel =0$. This implies $T{x}^{\ast}={x}^{\ast}$. So, ${x}^{\ast}$ is a fixed point of *T*.

*T*, then

we get $d({x}^{\ast},{y}^{\ast})\u2a7d0$. Thus $d({x}^{\ast},{y}^{\ast})=0$, which implies that ${x}^{\ast}={y}^{\ast}$, a contradiction. Hence, the fixed point is unique. □

**Remark 2.1** Note that in Theorem 2.1 we only suppose that the spectral radius of *k* is less than 1, neither $k<e$ nor $\parallel k\parallel <1$ assumed. This is a vital improvement. In fact, the condition $\rho (k)<1$ is weaker than that $\parallel k\parallel <1$, as is illustrated by Example 2.1 in the sequel. The improvement of the condition about the generalized Lipschitz constant *k* shows that it is meaningful to introduce the concepts of cone metric spaces with Banach algebras and a generalized Lipschitz condition.

**Theorem 2.2**

*Let*$(X,d)$

*be a complete cone metric space*,

*P*

*be a normal cone with a normal constant*

*M*.

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

*satisfies the generalized Lipschitz condition*

*where* $k\in P$ *with* $\rho (k)<\frac{1}{2}$. *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*Choose ${x}_{0}\in X$, and set ${x}_{n}={T}^{n}x$, $n\u2a7e1$. We have

which implies that $\{{x}_{n}\}$ is a Cauchy sequence by the proof of Theorem 2.1. Note that ${(e-k)}^{-1}$ and *k* commute.

*n*be large enough such that

where $\alpha \in \mathbb{R}$ such that ${lim}_{n\to \mathrm{\infty}}{\parallel {k}^{n}\parallel}^{\frac{1}{n}}<\alpha <\frac{1}{2}$.

where ${\beta}_{i}^{(n)}\in \mathbb{R}$, $n,i\u2a7e0$. It is easy to see that ${\beta}_{i}^{(n)}\u2a7e0$ for all $n,i\u2a7e0$.

and $\{{x}_{n}\}$ is a Cauchy sequence.

*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

*P*, we have

as $n\to \mathrm{\infty}$. Hence ${x}^{\ast}$ is a fixed point of *T*.

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

**Theorem 2.3**

*Let*$(X,d)$

*be a complete cone metric space*,

*P*

*be a normal cone with a normal constant*

*M*.

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

*satisfies the generalized Lipschitz condition*

*where* $k\in P$ *with* $\rho (k)<\frac{1}{2}$. *Then* *T* *has a unique fixed point in* *X*. *And for any* $x\in X$, *the iterative sequence* $\{{T}^{n}x\}$ *converges to the fixed point*.

*Proof*Choose ${x}_{0}\in X$, and set ${x}_{n}={T}^{n}x$, $n\u2a7e1$. We have

As is shown in the proof of Theorem 2.2, it follows that $\{{x}_{n}\}$ 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

as $n\to \mathrm{\infty}$.

*T*, then

which implies that ${x}^{\ast}={y}^{\ast}$, a contradiction. Hence, the fixed point is unique. □

We conclude the paper with an example.

**Example 2.1**Let $A={\mathbb{R}}^{2}$. For each $({x}_{1},{x}_{2})\in A$, $\parallel ({x}_{1},{x}_{2})\parallel =|{x}_{1}|+|{x}_{2}|$. The multiplication is defined by

Then *A* 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 a normal constant $M=1$.

*d*be defined by

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

where *α* can be any large positive real number.

Then, by Theorem 2.1, *T* has a unique fixed point in *X*.

**Remark 2.2** In Example 2.1 above, we see that $(\frac{1}{2},\alpha )\nless (1,0)=e$ and $\parallel (\frac{1}{2},\alpha )\parallel =\frac{1+2\alpha}{2}>1$ (for $\alpha >1$). Moreover, *T* is not a contractive mapping in the Euclidean metric on *X*. Hence, Example 2.1 shows that the main results in this paper are more powerful than the standard results of cone metric spaces presented in the literature.

**Remark 2.3**Example 2.1 also shows that one is unable to conclude that the cone metric space $(X,d)$ with a Banach algebra

*A*defined above is equivalent to the metric space $(X,{d}^{\ast})$, where the metric ${d}^{\ast}$ is defined by ${d}^{\ast}={\xi}_{e}\circ d$; here, the nonlinear scalarization function ${\xi}_{e}:A\to \mathbb{R}$ ($e\in intP$) is defined by

which implies that *T* is not a contraction in the metric space $(X,{d}^{\ast})$. This shows that one is unable to prove that Theorem 2.1 above is a consequence of the corresponding results in metric spaces by means of the methods 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 the PhD Start-up Fund of Hanshan Normal University, Guangdong Province, China (No. QD20110920).

## Authors’ Affiliations

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