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

*Fixed Point Theory and Applications*
**volume 2013**, Article number: 26 (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.

## 1 Introduction and preliminaries

The well-known Banach fixed point theorem asserts that if (X,d) is a complete metric space and T:X\to X is a map such that

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

On the contrary, assume 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.

Now, we show that \{{T}^{n}{x}_{0}\} is a Cauchy sequence. To prove the claim, we first show by induction that for each n\ge 2,

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

Thus, (2.2) holds for n=2. Suppose that (2.2) holds for each k<n, and we show that it holds for k=n. Since *T* is a generalized Ćirić quasi-contraction map, then we have

where

It is trivial that (2.2) holds if u=d({T}^{n-2}{x}_{0},{T}^{n-1}{x}_{0}). Now, suppose that u=d({T}^{n-2}{x}_{0},{T}^{n}{x}_{0}). In this case, we have

where

Again, it is trivial that (2.2) holds if {u}_{1}=d({T}^{n-1}{x}_{0},{T}^{n}{x}_{0}) or {u}_{1}=d({T}^{n-3}{x}_{0},{T}^{n-2}{x}_{0}). If {u}_{1}=d({T}^{n-2}{x}_{0},{T}^{n-1}{x}_{0}), then

By the assumption of induction,

Hence,

If {u}_{1}=d({T}^{n-3}{x}_{0},{T}^{n-1}{x}_{0}), then

If {u}_{1}=d({T}^{n-3}{x}_{0},{T}^{n}{x}_{0}), then

Therefore, by continuing this process, we see that (2.2) holds for each n\ge 2. From (2.2), we deduce that \{{T}^{n}{x}_{0}\} is a Cauchy sequence and since (X,d) is a generalized complete metric space, then there exists an \overline{x}\in X such that {lim}_{n\to \mathrm{\infty}}{T}^{n}{x}_{0}=\overline{x}. Now, we show that \overline{x} is a fixed point of *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

Then we have

which yields d(T\overline{x},\overline{x})=0, and so \overline{x}=T\overline{x} (note that 0<k<1 and d(T\overline{x},\overline{x})<\mathrm{\infty} by our assumptions). Now, let us assume that \overline{x} and \overline{y} are fixed points of *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

for all Sx,Sy\in SE. Since SE=SX is complete, by using Theorem 2.3, there exists \overline{x}\in X such that U(S\overline{x})=S\overline{x}. Then T\overline{x}=S\overline{x}, and so *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}. □

## References

Ćirić LB: A generalization of Banach’s contraction principle.

*Proc. Am. Math. Soc.*1974, 45(2):267–273.Amini-Harandi A: Fixed point theory for set-valued quasi-contraction maps in metric spaces.

*Appl. Math. Lett.*2011, 24: 1791–1794. 10.1016/j.aml.2011.04.033Sastry KPR, Naidu SVR: Fixed point theorems for generalized contraction mappings.

*Yokohama Math. J.*1980, 28: 15–19.Ćirić L, Hussain N, Cakić N: Common fixed points for Ćirić type

*f*-weak contraction with applications.*Publ. Math. (Debr.)*2010, 76: 31–49.Haghi RH, Rezapour S, Shahzad N: Some fixed point generalizations are not real generalizations.

*Nonlinear Anal.*2011, 74: 1799–1803. 10.1016/j.na.2010.10.052

## 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.

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Kiany, F., Amini-Harandi, A. Fixed point theory for generalized Ćirić quasi-contraction maps in metric spaces.
*Fixed Point Theory Appl* **2013**, 26 (2013). https://doi.org/10.1186/1687-1812-2013-26

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