# Suzuki-type fixed point results in metric type spaces

- Nawab Hussain
^{1}Email author, - Dragan Ðorić
^{2}, - Zoran Kadelburg
^{3}and - Stojan Radenović
^{4}

**2012**:126

https://doi.org/10.1186/1687-1812-2012-126

© Hussain et al.; licensee Springer 2012

**Received: **5 February 2012

**Accepted: **18 July 2012

**Published: **31 July 2012

## Abstract

Suzuki’s fixed point results from (Suzuki, Proc. Am. Math. Soc. 136:1861-1869, 2008) and (Suzuki, Nonlinear Anal. 71:5313-5317, 2009) are extended to the case of metric type spaces and cone metric type spaces. Examples are given to distinguish our results from the known ones.

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

## Keywords

## 1 Introduction and preliminaries

In 2008 Suzuki proved the following refinement of Banach’s fixed point principle.

**Theorem 1** ([1], Theorem 2])

*Let*$(X,d)$

*be a complete metric space*.

*Let*$T:X\to X$

*be a selfmap and*${\theta}_{1}:[0,1)\to (1/2,1]$

*be defined by*

*then* *T* *has a unique fixed point*$z\in X$*and for each*$x\in X$, *the sequence*$\{{T}^{n}x\}$*converges to z*.

There were various extensions of Suzuki’s result, such as Kikkawa-Suzuki’s version of Kannan’s theorem [2] and Popescu’s version of Ćirić’s theorem [3].

Suzuki proved also the following version of Edelstein’s fixed point theorem.

**Theorem 2** ([4], Theorem 3])

*Let*$(X,d)$

*be a compact metric space*.

*Let*$T:X\to X$

*be a selfmap*,

*satisfying for all*$x,y\in X$, $x\ne y$

*the condition*

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

This theorem was generalized in [5].

Let *E* be a real Banach space with the zero vector *θ*. A subset *P* of *E* is called a *cone* if: (a) *P* is closed, non-empty and $P\ne \{\theta \}$; (b) $a,b\in \mathbb{R}$, $a,b\ge 0$, $x,y\in P$ imply that $ax+by\in P$; (c) $P\cap (-P)=\{\theta \}$. Given a cone *P*, we define the partial ordering ⪯ with respect to *P* by $x\u2aafy$ if and only if $y-x\in P$. We shall write $x\ll y$ for $y-x\in intP$, where int*P* stands for the interior of *P* and use $x\prec y$ for $x\u2aafy$ and $x\ne y$. If $intP\ne \mathrm{\varnothing}$, then *P* is called a *solid cone*. It is said to be *normal* if there is a number $K>0$ such that for all $x,y\in E$, $\theta \u2aafx\u2aafy$ implies $\parallel x\parallel \le K\parallel y\parallel $. Such a minimal constant *K* is called the *normal constant* of *P*.

Huang and Zhang re-introduced cone metric spaces in [6] (this notion was known under various names since the mid of the 20th century, see a survey in [7]), replacing the set of real numbers by an ordered Banach space as the codomain for a metric. Cone metric spaces over normal cones inspired another generalization of metric spaces that were called *metric type spaces* by Khamsi [8] (see also [9–12]; note that, in fact, spaces of this kind were used earlier under the name of *b*-spaces by Czerwik [13]). Cvetković *et al.*[14] and Shah *et al.*[15] extended Khamsi’s definition and defined cone metric type spaces as follows:

*X*be a nonempty set,

*E*a Banach space with the solid cone

*P*and let $K\ge 1$ be a real number. If the function $D:X\times X\to P$ satisfies the following properties:

- (a)
$D(x,y)=0$ if and only if $x=y$;

- (b)
$D(x,y)=D(y,x)$ for all $x,y\in X$;

- (c)
$D(x,z)\u2aafK(D(x,y)+D(y,z))$ for all $x,y,z\in X$,

then *D* is called a *cone metric type function* and $(X,D,K)$ is called a *cone metric type space* (CMTS).

In particular, when $E=\mathbb{R}$ and $P=[0,+\mathrm{\infty})$, CMTS $(X,D,K)$ reduces to a *metric type space* (MTS) of [8, 9, 12].

Of course, for $K=1$ we get the *cone metric space* (CMS) of [6], resp. the usual metric space.

**Example 1** ([14])

*f*a.e. Further, let

It was shown in [14] that ${P}_{B}$ is a solid cone in ${\mathbb{R}}^{n}$ and that $({X}_{p},{D}_{p},{2}^{p-1})$ is a CMTS. In particular, for $n=1$ we get an MTS and for $p=1$ a CMS.

Let $(X,d)$ be any CMS over a normal cone with normal constant $K\ge 1$. Then $(X,D,K)$ is an MTS, where $D(x,y)=\parallel d(x,y)\parallel $. In this case the spaces $(X,d)$ and $(X,D,K)$ have the same topologies (see [10], Theorem 2.7]).

If $(X,{D}_{1},K)$ is a CMTS over a normal cone with a normal constant $k\ge 1$, then $(X,D,Kk)$ is an MTS, where $D(x,y)=\parallel {D}_{1}(x,y)\parallel $. Similarly as above, the spaces $(X,{D}_{1},K)$ and $(X,D,Kk)$ have the same topologies.

The last property always holds in the case of an MTS $(X,D,K)$ generated by a CMS $(X,d)$ over a normal cone, see Example 2, but not in general, as the following example shows.

**Example 3**Let $X=\mathbb{N}\cup \{\mathrm{\infty}\}$ and let $D:X\times X\to \mathbb{R}$ be defined by

that is, ${x}_{n}\to \mathrm{\infty}$, but $D({x}_{n},1)=2\nrightarrow D(\mathrm{\infty},1)$ as $n\to \mathrm{\infty}$.

Recall that a selfmap $T:X\to X$ is said to have the property (*P*) [16] if $F(T)=F({T}^{n})$ for each $n\in \mathbb{N}$, where $F(T)$ is the set of fixed points of *T*.

In this paper, we extend Suzuki’s Theorems 1 and 2, as well as Popescu’s results from [3] to the case of metric type spaces and cone metric type spaces. Examples are given to distinguish our results from the known ones.

## 2 Results

### 2.1 Results in metric type spaces

**Theorem 3**

*Let*$(X,D,K)$

*be a complete MTS where*

*D*

*is continuous in each variable*.

*Let*$T:X\to X$

*be a selfmap and*$\theta ={\theta}_{K}:[0,1)\to (1/(K+1),1]$

*be defined by*

*where*${b}_{K}=\frac{1-K+\sqrt{1+6K+{K}^{2}}}{4}$

*is the positive solution of*$\frac{1-r}{{r}^{2}}=\frac{1}{K+r}$.

*If there exists*$r\in [0,1)$

*such that for each*$x,y\in X$,

*where*

*then* *T* *has a unique fixed point*$z\in X$*and for each*$x\in X$, *the sequence*$\{{T}^{n}x\}$*converges to z*. *Moreover*, *T* *has the property* (*P*).

Note that for $K=1$, Theorem 3 reduces to a special case of Theorem 2.1 by Popescu [3].

*Proof*First note that $\theta (r)\le 1$ implies that $\theta (r)D(x,Tx)\le D(x,Tx)$ and it follows by (2.2) that

for each $x\in X$.

Using [12], Lemma 3.1] we conclude that $\{{u}_{n}\}$ is a Cauchy sequence, tending to some *z* in the complete space *X*. Obviously, also $T{u}_{n}={u}_{n+1}\to z$$n\to \mathrm{\infty}$.

*D*, $D({u}_{n},x)\to D(x,z)\ne 0$, it follows that there exists ${n}_{0}\in \mathbb{N}$ such that

*n*

*D*), we get that

It is easy to see that (2.6) follows from the previous relation.

and (2.9) is proved by induction.

In order to prove that $Tz=z$, we suppose that $Tz\ne z$ and consider the two possible cases.

It is easy to see (using (2.8), (2.9) and the inductive hypothesis) that the last maximum is equal to $D(z,Tz)$, *i.e.*, $D({T}^{n+1}z,Tz)\le \frac{r}{K}D(z,Tz)$ and relation (2.10) is proved by induction.

*n*, we have that

Thus, ${T}^{n}z\to z$ and, again from (2.10), we get that $D(Tz,z)\le \frac{r}{K}D(Tz,z)$ and $D(Tz,z)=0$, a contradiction.

*n*:

Passing to the limit when $j\to \mathrm{\infty}$ we get that $D(z,Tz)\le \frac{r}{K}D(z,Tz)$, which is possible only if $Tz=z$, a contradiction.

*z*is a fixed point of

*T*. The uniqueness of the fixed point follows easily from (2.6). Indeed, if

*y*

*z*are two fixed points of

*T*, then (2.6) implies that

wherefrom $y=z$. The property (*P*) follows from (2.3) (see [16]). □

Suzuki-Banach-type and Suzuki-Kannan-type fixed point results in metric type spaces (versions of [1], Theorem 2] and [2], Theorem 2.2]) are special cases of Theorem 3.

**Corollary 1**

*Let*$(X,D,K)$

*be a complete MTS where*

*D*

*is continuous in each variable*.

*Let*$T:X\to X$

*be a selfmap and*$\theta :[0,1)\to (1/(K+1),1]$

*be defined by*(2.1).

*If there exists*$r\in [0,1)$

*such that for each*$x,y\in X$,

*then* *T* *has a unique fixed point*$z\in X$*and for each*$x\in X$, *the sequence*$\{{T}^{n}x\}$*converges to z*. *Moreover*, *T* *has the property* (*P*).

**Corollary 2**

*Let*$(X,D,K)$

*be a complete MTS where*

*D*

*is continuous in each variable*.

*Let*$T:X\to X$

*be a selfmap and*$\theta :[0,1)\to (1/(K+1),1]$

*be defined by*(2.1).

*If there exists*$r\in [0,1)$

*such that for each*$x,y\in X$,

*then* *T* *has a unique fixed point*$z\in X$*and for each*$x\in X$, *the sequence*$\{{T}^{n}x\}$*converges to z*. *Moreover*, *T* *has the property* (*P*).

**Corollary 3**

*Let*$(X,D,K)$

*be a complete MTS where*

*D*

*is continuous in each variable*.

*Let*$T:X\to X$

*be a selfmap and*$\theta :[0,1)\to (1/(K+1),1]$

*be defined by*(2.1).

*If there exists*$r\in [0,1)$

*such that for each*$x,y\in X$,

*then* *T* *has a unique fixed point*$z\in X$*and for each*$x\in X$, *the sequence*$\{{T}^{n}x\}$*converges to z*. *Moreover*, *T* *has the property* (*P*).

Adapting [1], Example 1] we give now an example of a mapping satisfying the conditions of Theorem 3 (and having a unique fixed point) but not satisfying the respective classical (non-Suzuki-type) condition in metric type spaces (see, *e.g.*, [14], Theorem 3.4]).

**Example 4**Let $X=\{(0,0),(4,0),(0,4),(4,6),(6,4)\}$, and let $D:X\times X\to [0,+\mathrm{\infty})$ be given by $D(({x}_{1},{x}_{2}),({y}_{1},{y}_{2}))={({x}_{1}-{y}_{1})}^{2}+{({x}_{2}-{y}_{2})}^{2}$. Then $(X,D,2)$ is a metric type space (see Example 1). Let $T:X\to X$ be given as

Let now $x,y\in \{(4,6),(6,4)\}$, $x\ne y$. Then $D(x,y)=8$ and $D(x,Tx)=36$ and so $\theta (r)D(x,Tx)>\frac{1}{3}\cdot 36>8=D(x,y)$, and (2.2) is trivially satisfied. Note that in the classical variant, in this case $D(Tx,Ty)=32$ and $M(x,y)=36$, so the inequality $D(Tx,Ty)\le \frac{r}{2}M(x,y)$ does not hold for any $r<1$.

The following is a metric-type version of Theorem 2.

**Theorem 4**

*Let*$(X,D,K)$

*be a compact MTS*,

*where the function*

*D*

*is continuous*.

*Let*$T:X\to X$

*be a selfmap*,

*satisfying for all*$x,y\in X$, $x\ne y$

*the condition*

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

*Proof* Denote $\beta =inf\{D(x,Tx):x\in X\}$ and choose a sequence $\{{x}_{n}\}$ in *X* such that $D({x}_{n},T{x}_{n})\to \beta $ ($n\to \mathrm{\infty}$). Since the space *X* is (sequentially) compact, we can suppose that there exist $v,w\in X$ such that ${x}_{n}\to v$ and $T{x}_{n}\to w$ ($n\to \mathrm{\infty}$). We will prove that $\beta =0$.

*D*implies that ${lim}_{n\to \mathrm{\infty}}D({x}_{n},w)=D(v,w)={lim}_{n\to \mathrm{\infty}}D({x}_{n},T{x}_{n})=\beta $. Choose ${n}_{0}\in \mathbb{N}$ such that for all $n\ge {n}_{0}$

holds true. Then $\frac{1}{1+K}D({x}_{n},T{x}_{n})<D({x}_{n},w)$ and assumption (2.13) implies that $D(T{x}_{n},Tw)<\frac{1}{K}D({x}_{n},w)$ for $n\ge {n}_{0}$. Passing to the limit, we obtain that $D(w,Tw)\le \frac{1}{K}\beta $. If $K>1$, the last inequality is impossible by the definition of *β*. If $K=1$, it is possible only if $D(w,Tw)=\beta $ (recall that we have supposed that $\beta >0$). But in this case $\frac{1}{1+K}D(w,Tw)<D(w,Tw)$ and (2.13) implies that $D(Tw,{T}^{2}w)<\frac{1}{K}D(w,Tw)=\beta $, which is again impossible by the definition of *β*. Hence, in all cases we obtain a contradiction and it follows that $\beta =0$ and so $v=w$.

*T*has a fixed point, suppose that $Tz\ne z$ for all $z\in X$. Then, in particular, $0<\frac{1}{1+K}D({x}_{n},T{x}_{n})<D({x}_{n},T{x}_{n})$ and (2.13) implies that

when $n\to \mathrm{\infty}$. Hence, ${T}^{2}{x}_{n}\to v$ ($n\to \mathrm{\infty}$).

holds. In other words, there exists a sequence $\{{n}_{j}\}$ such that $D(T{x}_{{n}_{j}},Tv)<\frac{1}{K}D({x}_{{n}_{j}},v)$ holds for each $j\in \mathbb{N}$, or there exists a sequence $\{{n}_{k}\}$ such that $D({T}^{2}{x}_{{n}_{k}},Tv)<\frac{1}{K}D({x}_{{n}_{k}},v)$ holds for each $k\in \mathbb{N}$. In both cases, passing to the limit, we obtain that $D(v,Tv)=0$, *i.e.*, $Tv=v$, a contradiction with the assumption that *T* has no fixed points.

It follows that there exists $z\in X$ such that $Tz=z$. Uniqueness follows easily. □

### 2.2 Results in cone metric type spaces

In this subsection, we formulate cone-metric-type versions of the results from the previous subsection.

**Theorem 5**

*Let*$(X,{D}_{1},K)$

*be a complete CMTS with the normal underlying cone*

*P*,

*where*${D}_{1}$

*is continuous in each variable*.

*Let*$T:X\to X$

*be a selfmap and*$\theta ={\theta}_{K}:[0,1)\to (1/(K+1),1]$

*be defined by*(2.1).

*If there exists*$r\in [0,1)$

*such that for each*$x,y\in X$,

*for some*

*then* *T* *has a unique fixed point*$z\in X$*and for each*$x\in X$, *the sequence*$\{{T}^{n}x\}$*converges to z*.

*Proof*Since the cone

*P*is normal, without loss of generality, we can assume that the normal constant of

*P*is $k=1$ and that the given norm in

*E*is monotone,

*i.e.*(see [17], Lemma 2.1]). Denote $D(x,y)=\parallel {D}_{1}(x,y)\parallel $. Then

*D*is a (real-valued) metric-type function and the space $(X,D)$ is compact (together with $(X,{D}_{1})$, see [10], Theorem 2.7]). Let us prove that the mapping

*T*satisfies for some $r\in [0,1)$ the condition

*i.e.*$\theta (r){D}_{1}(x,Tx)\gg {D}_{1}(x,y)$, it would follow that $\theta (r)D(x,Tx)>D(x,y)$, a contradiction with the assumption). Assumption (2.14) implies that ${D}_{1}(Tx,Ty)\u2aaf\frac{r}{K}u(x,y)$ for some

Hence, condition (2.15) is satisfied, and the conclusion follows. □

In a similar way, the following corollaries and the theorem can be proved.

**Corollary 4**

*Let*$(X,{D}_{1},K)$

*be a complete CMTS where*${D}_{1}$

*is continuous in each variable*.

*Let*$T:X\to X$

*be a selfmap and*$\theta :[0,1)\to (1/(K+1),1]$

*be defined by*(2.1).

*If there exists*$r\in [0,1)$

*such that for each*$x,y\in X$,

*then* *T* *has a unique fixed point*$z\in X$*and for each*$x\in X$, *the sequence*$\{{T}^{n}x\}$*converges to z*.

**Corollary 5**

*Let*$(X,{D}_{1},K)$

*be a complete CMTS where*${D}_{1}$

*is continuous in each variable*.

*Let*$T:X\to X$

*be a selfmap and*$\theta :[0,1)\to (1/(K+1),1]$

*be defined by*(2.1).

*If there exists*$r\in [0,1)$

*such that for each*$x,y\in X$,

*where*$u(x,y)\in \{{D}_{1}(x,Tx),{D}_{1}(y,Ty)\}$, *then* *T* *has a unique fixed point*$z\in X$*and for each*$x\in X$, *the sequence*$\{{T}^{n}x\}$*converges to* *z*.

**Corollary 6**

*Let*$(X,{D}_{1},K)$

*be a complete CMTS where*${D}_{1}$

*is continuous in each variable*.

*Let*$T:X\to X$

*be a selfmap and*$\theta :[0,1)\to (1/(K+1),1]$

*be defined by*(2.1).

*If there exists*$r\in [0,1)$

*such that for each*$x,y\in X$,

*then* *T* *has a unique fixed point*$z\in X$*and for each*$x\in X$, *the sequence*$\{{T}^{n}x\}$*converges to z*.

Example 4 can be easily adapted to a CMTS.

**Theorem 6**

*Let*$(X,{D}_{1},K)$

*be a compact CMTS*,

*where the function*${D}_{1}$

*is continuous*.

*Let*$T:X\to X$

*be a selfmap satisfying*,

*for all*$x,y\in X$, $x\ne y$

*the condition*

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

Note that for $K=1$ the above theorem reduces to [5], Theorem 3.8].

## Declarations

### Acknowledgements

The first author gratefully acknowledges the support provided by the Deanship of Scientific Research (DSR), King Abdulaziz University during this research. The second, third and fourth authors are thankful to the Ministry of Science and Technological Development of Serbia.

## Authors’ Affiliations

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