# Fixed point theory for generalized Ćirić quasi-contraction maps in metric spaces

- F Kiany
^{1}and - A Amini-Harandi
^{2, 3}Email author

**2013**:26

https://doi.org/10.1186/1687-1812-2013-26

© Kiany and Amini-Harandi; licensee Springer 2013

**Received: **16 June 2012

**Accepted: **13 January 2013

**Published: **6 February 2013

## Abstract

In this paper, we first give a new fixed point theorem for generalized Ćirić quasi-contraction maps in generalized metric spaces. Then we derive a common fixed point result for quasi-contractive type maps. Some examples are given to support our results. Our results extend and improve some fixed point and common fixed point theorems in the literature.

**MSC:**47H10.

## Keywords

## 1 Introduction and preliminaries

where $0\le c<1$, then *T* has a unique fixed point $\overline{x}\in X$ and for any ${x}_{0}\in X$, the sequence $\{{T}^{n}{x}_{0}\}$ converges to $\overline{x}$.

In recent years, a number of generalizations of the above Banach contraction principle have appeared. Of all these, the following generalization of Ćirić [1] stands at the top.

**Theorem 1.1**

*Let*$(X,d)$

*be a complete metric space*.

*Let*$T:X\to X$

*be a Ćirić quasi*-

*contraction map*,

*that is*,

*there exists*$c<1$

*such that*

*for any* $x,y\in X$. *Then* *T* *has a unique fixed point* $\overline{x}\in X$ *and for any* ${x}_{0}\in X$, *the sequence* $\{{T}^{n}{x}_{0}\}$ *converges to* $\overline{x}$.

For other generalizations of the above theorem, see [2] and the references therein.

## 2 Main results

Let *X* be a nonempty set and let $d:X\times X\to [0,\mathrm{\infty}]$ be a mapping. If *d* satisfies all of the usual conditions of a metric except that the value of *d* may be infinity, we say that $(X,d)$ is a *generalized metric space*.

We now introduce the concept of a *generalized Ćirić quasi-contraction* map in generalized metric spaces.

**Definition 2.1**Let $(X,d)$ be a generalized metric space. The self-map $T:X\to X$ is said to be a

*generalized Ćirić quasi-contraction*if

for any $x,y\in X$, where $\alpha :[0,\mathrm{\infty}]\to [0,1)$ is a mapping.

As the following simple example due to Sastry and Naidu [3] shows, Theorem 1.1 is not true for generalized Ćirić quasi-contraction maps even if we suppose *α* is continuous and increasing.

**Example 2.2**Let $X=[1,\mathrm{\infty})$ with the usual metric, $T:X\to X$ be given by $Tx=2x$. Define $\alpha :[0,\mathrm{\infty})\to [0,1)$ by $\alpha (t)=\frac{2t}{1+2t}$. Then, clearly,

*α*is continuous and increasing, and

for each $x,y\in X$, but *T* has no fixed point.

Now, a natural question is what further conditions are to be imposed on *T* or *α* to guarantee the existence of a fixed point for *T*? For some partial answers to this question and application of quasi-contraction maps to variational inequalities, see [4] and the references therein.

Now, we are ready to state our main result.

**Theorem 2.3**

*Let*$(X,d)$

*be a complete generalized metric space*.

*Let*$T:X\to X$

*be a generalized Ćirić quasi*-

*contraction map such that*

*α*

*satisfies*

*Assume that there exists an* ${x}_{0}\in X$ *with the bounded orbit*, *that is*, *the sequence* $\{{T}^{n}{x}_{0}\}$ *is bounded*. *Furthermore*, *suppose that* $d(x,Tx)<\mathrm{\infty}$ *for each* $x\in X$. *Then* *T* *has a fixed point* $\overline{x}\in X$ *and* ${lim}_{n\to \mathrm{\infty}}{T}^{n}{x}_{0}=\overline{x}$. *Moreover*, *if* $\overline{y}$ *is a fixed point of* *T*, *then either* $d(\overline{x},\overline{y})=\mathrm{\infty}$ *or* $\overline{x}=\overline{y}$.

*Proof*If for some ${n}_{0}\in \mathbb{N}$, ${T}^{{n}_{0}-1}{x}_{0}={T}^{{n}_{0}}x=T({T}^{{n}_{0}-1}{x}_{0})$, then ${T}^{n}{x}_{0}={T}^{{n}_{0}-1}{x}_{0}$ for $n\ge {n}_{0}$. Thus, ${T}^{{n}_{0}-1}{x}_{0}$ is a fixed point of

*T*, the sequence $\{{T}^{n}{x}_{0}\}$ is convergent to ${T}^{{n}_{0}-1}{x}_{0}$, and we are finished (note that ${T}^{n}{x}_{0}={T}^{{n}_{0}-1}{x}_{0}$ for each $n\ge {n}_{0}$). So, we may assume that ${T}^{n-1}{x}_{0}\ne {T}^{n}{x}_{0}$ for each $n\in \mathbb{N}$. Now, we show that there exists $0<c<1$ such that

for some subsequence $\{\alpha (d({T}^{{n}_{k}-1}{x}_{0},{T}^{{n}_{k}}{x}_{0}))\}$ of $\{\alpha (d({T}^{n-1}{x}_{0},{T}^{n}{x}_{0}))\}$. Since by our assumption the sequence $\{d({T}^{n-1}{x}_{0},{T}^{n}{x}_{0})\}$ is bounded, then the subsequence $\{d({T}^{{n}_{k}-1}{x}_{0},{T}^{{n}_{k}}{x}_{0})\}$ is bounded too, and so, by passing to subsequences if necessary, we may assume that it is convergent. Let ${r}_{0}={lim}_{k\to \mathrm{\infty}}d({T}^{{n}_{k}-1}{x}_{0},{T}^{{n}_{k}}{x}_{0})$. Then from (2.1), we have ${lim\hspace{0.17em}sup}_{t\to {r}_{0}}\alpha (t)=1$, a contradiction. Thus, (2.1) holds.

*K*is a bound for the bounded sequence ${\{d({x}_{0},{T}^{n}{x}_{0})\}}_{n}$. If $n=2$ then, we get

*T*is a generalized Ćirić quasi-contraction map, then we have

*T*. To show the claim, we first show that there exists $0<k<1$ such that $\alpha (d(\overline{x},{T}^{n}{x}_{0}))<k$ for each $n\in \mathbb{N}$. On the contrary, assume that ${lim}_{j\to \mathrm{\infty}}\alpha (d(\overline{x},{T}^{{n}_{j}}{x}_{0}))=1$ for some subsequence ${n}_{j}$. Since ${lim}_{j\to \mathrm{\infty}}d(\overline{x},{T}^{{n}_{j}}{x}_{0})=0$, then from the above, we get ${lim\hspace{0.17em}sup}_{t\to {0}^{+}}\alpha (t)=1$, a contradiction. Since

*T*is a generalized Ćirić quasi-contraction, then we have

*T*such that $d(\overline{x},\overline{y})<\mathrm{\infty}$. Then

and so $\overline{x}=\overline{y}$ (note that $\alpha (d(\overline{x},\overline{y}))<1$). □

The following example shows that in the statement of Theorem 2.3, the condition $d(x,Tx)<\mathrm{\infty}$ for each $x\in X$ is necessary.

**Example 2.4**Let $X=\{0,\mathrm{\infty}\}$, $d(0,0)=d(\mathrm{\infty},\mathrm{\infty})=0$ and let $d(0,\mathrm{\infty})=\mathrm{\infty}$. Let $T:X\to X$ be given by $T0=\mathrm{\infty}$ and $T\mathrm{\infty}=0$. Then

for each $x,y\in X$, but *T* is fixed point free.

**Example 2.5**Let $X=[0,\mathrm{\infty}]$, $d(x,y)=|x-y|$ for each $x,y\in [0,\mathrm{\infty})$, $d(x,\mathrm{\infty})=\mathrm{\infty}$ for each $x\in [0,\mathrm{\infty})$ and let $d(\mathrm{\infty},\mathrm{\infty})=0$. Then $(X,d)$ is a complete generalized metric space. Let $T:X\to X$ be given by $Tx=2x$ for each $x\in [0,\mathrm{\infty})$ and $T\mathrm{\infty}=\mathrm{\infty}$. Define $\alpha :[0,\mathrm{\infty}]\to [0,1)$ by $\alpha (t)=\frac{2t}{1+2t}$ for each $t\in [0,\mathrm{\infty})$ and $\alpha (\mathrm{\infty})=\frac{1}{2}$. Then we have

and $d(x,Tx)<\mathrm{\infty}$ for each $x,y\in X$. Thus, all of the assumptions of Theorem 2.3 are satisfied, and so *T* has a unique fixed point ($x=\mathrm{\infty}$ is a unique fixed point of *T*). But we cannot invoke the above mentioned theorem of Ćirić to show the existence of a fixed point for *T*.

To prove the following common fixed point result, we use the technique in [5].

**Corollary 2.6**

*Let*$(X,d)$

*be a complete metric space and let the self*-

*maps*

*T*

*and*

*S*

*satisfy the contractive condition*

*for each* $x,y\in X$, *where* *α* *satisfies* ${lim\hspace{0.17em}sup}_{t\to {r}^{+}}\alpha (t)<1$ *for each* $r\in [0,\mathrm{\infty})$. *If* $TX\subseteq SX$ *and* *SX* *is a complete subset of* *X*, *then* *T* *and* *S* *have a unique coincidence point in* *X*. *Moreover*, *if* *T* *and* *S* *are weakly compatible* (*i*.*e*., *they commute at their coincidence points*), *then* *T* *and* *S* *have a unique common fixed point*.

*Proof*It is well known that there exists $E\subseteq X$ such that $SE=SX$ and $S:E\to X$ is one-to-one. Now, define a map $U:SE\to SE$ by $U(Sx)=Tx$. Since

*S*is one-to-one on

*E*,

*U*is well defined. Note that

*T*and

*S*have a coincidence point, which is also unique. Since $T\overline{x}=S\overline{x}$ and

*T*and

*S*commute, then we have

Thus, $T\overline{x}=S\overline{x}$ is also a coincidence point of *T* and *S*. By the uniqueness of a coincidence point of *T* and *S*, we get $T\overline{x}=S\overline{x}=\overline{x}$. □

## Declarations

### Acknowledgements

The authors are grateful to the referees for their helpful comments leading to improvement of the presentation of the work. The second author acknowledge that this research was partially carried out at IPM-Isfahan Branch. The second author was partially supported by a grant from IPM (No. 91470412) and by the Center of Excellence for Mathematics, University of Shahrekord, Iran.

## Authors’ Affiliations

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