# Fixed and periodic points of generalized contractions in metric spaces

- Mujahid Abbas
^{1}, - Basit Ali
^{2}and - Salvador Romaguera
^{3}Email author

**2013**:243

https://doi.org/10.1186/1687-1812-2013-243

© Abbas et al.; licensee Springer. 2013

**Received: **1 July 2013

**Accepted: **8 October 2013

**Published: **7 November 2013

## Abstract

Wardowski (Fixed Point Theory Appl. 2012:94, 2012, doi:10.1186/1687-1812-2012-94) introduced a new type of contraction called *F*-contraction and proved a fixed point result in complete metric spaces, which in turn generalizes the Banach contraction principle. The aim of this paper is to introduce *F*-contractions with respect to a self-mapping on a metric space and to obtain common fixed point results. Examples are provided to support results and concepts presented herein. As an application of our results, periodic point results for the *F*-contractions in metric spaces are proved.

**MSC:**47H10, 47H07, 54H25.

## Keywords

*F*-contractionproperty

*P*property

*Q*common fixed point

## 1 Introduction and preliminaries

The Banach contraction principle [1] is a popular tool in solving existence problems in many branches of mathematics (see, *e.g.*, [2–4]). Extensions of this principle were obtained either by generalizing the domain of the mapping or by extending the contractive condition on the mappings [5–9]. Initially, existence of fixed points in ordered metric spaces was investigated and applied by Ran and Reurings [10]. Since then, a number of results have been proved in the framework of ordered metric spaces (see [11–18]). Contractive conditions involving a pair of mappings are further additions to the metric fixed point theory and its applications (for details, see [19–23]).

Recently, Wardowski [24] introduced a new contraction called *F*-contraction and proved a fixed point result as a generalization of the Banach contraction principle [1]. In this paper, we introduce an *F*-contraction with respect to a self-mapping on a metric space and obtain common fixed point results in an ordered metric space. In the last section, we give some results on periodic point properties of a mapping and a pair of mappings in a metric space. We begin with some basic known definitions and results which will be used in the sequel. Throughout this article, ℕ, ${\mathbb{R}}_{+}$, ℝ denote the set of natural numbers, the set of positive real numbers and the set of real numbers, respectively.

**Definition 1** Let *f* and *g* be self-mappings on a set *X*. If $fx=gx=w$ for some *x* in *X*, then *x* is called a coincidence point of *f* and *g* and *w* is called a coincidence point of *f* and *g*. Furthermore, if $fgx=gfx$ whenever *x* is a coincidence point of *f* and *g*, then *f* and *g* are called weakly compatible mappings [22].

Let $C(f,g)=\{x\in X:fx=gx\}$ ($F(f,g)=\{x\in X:x=fx=gx\}$) denote the set of all coincidence points (the set of all common fixed points) of self-mappings *f* and *g*.

**Definition 2** ([25])

*f*is called a

*g*-contraction if there exists $\alpha \in (0,1)$ such that

holds for all $x,y\in X$.

In 1976, Jungck [25] obtained the following useful generalization of the Banach contraction principle.

**Theorem 1** *Let* *g* *be a continuous self*-*mapping on a complete metric space* $(X,d)$. *Then* *g* *has a fixed point in* *X* *if and only if there exists a* *g*-*contraction mapping* $f:X\to X$ *such that* *f* *commutes with* *g* *and* $g(X)\subseteq f(X)$.

*Ϝ*be the collection of all mappings $F:{\mathbb{R}}_{+}\to \mathbb{R}$ that satisfy the following conditions:

- (C1)
*F*is strictly increasing, that is, for all $\alpha ,\beta \in {\mathbb{R}}_{+}$ such that $\alpha <\beta $ implies that $F(\alpha )<F(\beta )$. - (C2)
For every sequence ${\{{\alpha}_{n}\}}_{n\in \mathbb{N}}$ of positive real numbers, ${lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0$ and ${lim}_{n\to \mathrm{\infty}}F({\alpha}_{n})=-\mathrm{\infty}$ are equivalent.

- (C3)There exists $k\in (0,1)$ such that$\underset{\alpha \to {0}^{+}}{lim}{\alpha}^{k}F(\alpha )=0.$

**Definition 3** ([24])

*F*-contraction on

*X*if there exists $\tau >0$ such that

for all $x,y\in X$.

Note that every *F*-contraction is continuous (see [24]). We extend the above definition to two mappings.

**Definition 4**Let $(X,d)$ be a metric space, $F\in \u03dc$ and $f,g:X\to X$. The mapping

*f*is said to be an

*F*-contraction with respect to

*g*on

*X*if there exists $\tau >0$ such that

for all $x,y\in X$ satisfying $min\{d(fx,fy),d(gx,gy)\}>0$.

By different choices of mappings *F* in (1) and (2), one obtains a variety of contractions [24].

**Example 1**Let ${F}_{1}:{\mathbb{R}}_{+}\to \mathbb{R}$ be given by ${F}_{1}(\alpha )=ln(\alpha )$. It is clear that $F\in \u03dc$. Suppose that $f:X\to X$ is an

*F*-contraction with respect to a self-mapping

*g*on

*X*. From (2) we have

Therefore an ${F}_{1}$-contraction map *f* with respect to *g* reduces to a *g*-contraction mapping.

Now we give an example of an *F*-contraction with respect to a self-mapping *g* on *X* which is not a *g*-contraction on *X*.

**Example 2**Consider the following sequence of partial sums ${\{{S}_{n}\}}_{n\in \mathbb{N}}$ [[24], Example 2.5]:

*d*be the usual metric on

*X*. Let $f:X\to X$ and $g:X\to X$ be defined as

*f*is not a

*g*-contraction. If we take ${F}_{2}(\alpha )=ln(\alpha )+\alpha $, then ${F}_{2}\in \u03dc$ and

*f*is an ${F}_{2}$-contraction with respect to a mapping

*g*(taking $\tau =2$). Indeed, the following holds:

**Definition 5** ([26], Dominance condition)

Let $(X,\u2aaf)$ be a partially ordered set. A self-mapping *f* on *X* is said to be (i) a dominated map if $fx\u2aafx$ for each *x* in *X*, (ii) a dominating map if $x\u2aaffx$ for each *x* in *X*.

**Example 3**Let $X=[0,1]$ be endowed with the usual ordering and $f,g:X\to X$ defined by $gx={x}^{n}$ for some $n\in \mathbb{N}$ and $fx=kx$ for some real number $k\ge 1$. Note that

for all *x* in *X*. Thus *g* is dominated and *f* is a dominating map.

**Definition 6** Let $(X,\u2aaf)$ be a partially ordered set. Two mappings $f,g:X\to X$ are said to be weakly increasing if $fx\u2aafgfx$ and $gx\u2aaffgx$ for all *x* in *X* (see [27]).

**Definition 7** Let *X* be a nonempty set. Then $(X,d,\u2aaf)$ is called an ordered metric space if $(X,d)$ is a metric space and $(X,\u2aaf)$ is a partially ordered set.

**Definition 8**Let $(X,\u2aaf)$ be a partial ordered set, then

*x*,

*y*in

*X*are called comparable elements if either $x\u2aafy$ or $y\u2aafx$ holds true. Moreover, we define $\mathrm{\Delta}\subseteq X\times X$ by

**Definition 9** An ordered metric space $(X,d,\u2aaf)$ is said to have the sequential limit comparison property if for every non-decreasing sequence (non-increasing sequence) ${\{{x}_{n}\}}_{n\in \mathbb{N}}$ in *X* such that ${x}_{n}\to x$ implies that ${x}_{n}\u2aafx$ ($x\u2aaf{x}_{n}$).

## 2 Common fixed point results in ordered metric spaces

We present the following theorem as a generalization of results in [25] and [[24], Theorem 2.1].

**Theorem 2**

*Let*$(X,\u2aaf)$

*be a partially ordered set such that there exists a metric*

*d*

*on*

*X*,

*and let*$f:X\to X$

*be an*

*F*-

*contraction with respect to*$g:X\to X$

*on*Δ

*with*$f(X)\subseteq g(X)$.

*Assume that*

*f*

*is dominating and*

*g*

*is dominated*.

*Then*

- (a)
*f**and**g**have a coincidence point in**X**provided that*$g(X)$*is complete and has the sequential limit comparison property*. - (b)
$C(f,g)$

*is well ordered if and only if*$C(f,g)$*is a singleton*. - (c)
*f**and**g**have a unique common fixed point if**f**and**g**are weakly compatible and*$C(f,g)$*is well ordered*.

*Proof*(a) Let ${x}_{0}$ be an arbitrary point of

*X*. Since the range of

*g*contains the range of

*f*, there exists a point ${x}_{1}$ in

*X*such that $f({x}_{0})=g({x}_{1})$. As

*f*is dominating and

*g*is dominated, so we have

*X*, we obtain ${x}_{n+1}$ in

*X*such that

*f*is an

*F*-contraction with respect to

*g*on Δ, so we obtain

*q*in $g(X)$ such that ${lim}_{n\to \mathrm{\infty}}g{x}_{n}=q$. Let $p\in X$ be such that $g(p)=q$. The sequential limit comparison property implies that $g{x}_{n+1}\u2aafq$. As ${x}_{n}\u2aaff{x}_{n}=g{x}_{n+1}\u2aafq=g(p)\u2aafp$ so $({x}_{n},p)\in \mathrm{\Delta}$. Hence from (2) we have

Since ${lim}_{n\to \mathrm{\infty}}d(g{x}_{n-1},gp)=0$, therefore by (C2) we have ${lim}_{n\to \mathrm{\infty}}F(d(g{x}_{n-1},gp))=-\mathrm{\infty}$. Hence ${lim}_{n\to \mathrm{\infty}}F(d(g{x}_{n},fp))=-\mathrm{\infty}$ implies that ${lim}_{n\to \mathrm{\infty}}d(g{x}_{n},fp)=0$. That is, ${lim}_{n\to \mathrm{\infty}}g{x}_{n}=fp$. Uniqueness of limit implies $fp=gp$, that is, $p\in C(f,g)$.

*w*in

*X*such that $fw=gw$ with $w\ne p$. Since $C(f,g)$ is well ordered, so $(w,p)\in \mathrm{\Delta}$. Now from (2) we have

*f*and

*g*have a unique coincidence point

*p*in

*X*. The converse follows immediately.

- (c)
Now if

*f*and*g*are weakly compatible mappings, then we have $fq=fgp=gfp=gq$, that is,*q*is the coincidence point of*f*and*g*. But*q*is the only point of coincidence of*f*and*g*, so $fq=gq=q$. Hence*q*is the unique common fixed point of*f*and*g*. □

**Example 4**Let $X=[0,5]$ be endowed with usual metric and usual order. Define mappings $f,g:X\to X$ by

*g*is dominated and

*f*is dominating. Define $F:{\mathbb{R}}_{+}\to \mathbb{R}$ as $F(x)=ln(x)$. If $x\in [0,3)$ and $y\in [3,5)$, then

is satisfied for all $x,y\in [0,5]$, whenever $min\{d(fx,fy),d(gx,gy)\}>0$. Hence *f* is an *F*-contraction with respect to *g* on $[0,5]$. Hence all the conditions of Theorem 2 are satisfied. Moreover, $x=5$ is the coincidence point of *f* and *g*. Also note that *f* and *g* are weakly compatible and $x=5$ is the common fixed point of *g* and *f* as well.

Now we give a common fixed point result without imposing any type of commutativity condition for self-mappings *f* and *g* on *X*. Moreover, we relax the dominance conditions on *f* and *g* as well.

**Theorem 3**

*Let*$(X,\u2aaf)$

*be a partially ordered set such that there exists a complete metric*

*d*

*on*

*X*.

*If self*-

*mappings*

*f*

*and*

*g*

*on*

*X*

*are weakly increasing and for some*$\tau >0$

*satisfy*

*for all* $(x,y)\in \mathrm{\Delta}$ *such that* $min\{d(fx,gy),d(x,y)\}>0$, *then* $F(f,g)\ne \mathrm{\varnothing}$, *provided that* *X* *has the sequential limit comparison property*. *Further*, *f* *and* *g* *have a unique common fixed point if and only if* $F(f,g)$ *is well ordered*.

*Proof*Let ${x}_{0}$ be an arbitrary point of

*X*. Define a sequence ${\{{x}_{n}\}}_{n\in \mathbb{N}}$ in

*X*as follows: ${x}_{2n+1}=f{x}_{2n}$ and ${x}_{2n+2}=g{x}_{2n+1}$. Since

*f*and

*g*are weakly increasing, we have ${x}_{2n+1}=f{x}_{2n}\u2aafgf{x}_{2n}=g{x}_{2n+1}={x}_{2n+2}$ and ${x}_{2n+2}=g{x}_{2n+1}\u2aaffg{x}_{2n+1}=f{x}_{2n+2}={x}_{2n+3}$. Hence $({x}_{2n+1},{x}_{2n+2})\in \mathrm{\Delta}$ and $({x}_{2n+2},{x}_{2n+3})\in \mathrm{\Delta}$ for every $n\in \mathbb{N}\cup \{0\}$. Now define

*X*, so there exists

*p*in

*X*such that ${lim}_{n\to \mathrm{\infty}}{x}_{n}=p$. As

*X*has the sequential limit comparison property, so $({x}_{n},p),({x}_{2n},p),({x}_{2n+1},p)\in \mathrm{\Delta}$. Therefore

Since ${lim}_{n\to \mathrm{\infty}}d({x}_{2n},p)=0$, by (C2) we have ${lim}_{n\to \mathrm{\infty}}F(d({x}_{2n},p))=-\mathrm{\infty}$. This implies ${lim}_{n\to \mathrm{\infty}}F(d({x}_{2n+1},gp))=-\mathrm{\infty}$, which further implies that ${lim}_{n\to \mathrm{\infty}}d({x}_{2n+1},gp)=0$. Hence $d(p,gp)=0$ and $p=gp$. Similarly, we obtain $p=fp$. This shows that *p* is a common fixed point of *g* and *f*. Now suppose that $F(f,g)$ is well ordered. We prove that $F(f,g)$ is a singleton. Assume on the contrary that there exists another point *q* in *X* such that $q=fq=gq$ with $q\ne p$. Obviously, $(q,p)\in \mathrm{\Delta}$. So, from (5) we have $\tau \le F(d(q,p))-F(d(fq,gp))=0$, a contradiction. Therefore $q=p$. Hence *g* and *f* have a unique common fixed point *p* in *X*. The converse follows immediately. □

## 3 Periodic point results in metric spaces

If *x* is a fixed point of the self-mapping *f*, then *x* is a fixed point of ${f}^{n}$ for every $n\in \mathbb{N}$, but the converse is not true. In the sequel, we denote by $F(f)$ the set of all fixed points of *f*.

**Example 5**Let $f:[0,1]\to [0,1]$ be given by

*f*has a unique fixed point $x=1/2$. Note that ${f}^{n}x=x$ holds for every even natural number

*n*and

*x*in $[0,1]$. On the other hand, define a mapping $g:[0,\pi ]\to [0,\pi ]$ as

Then *g* has the same fixed point as ${g}^{n}$ for every *n*.

**Definition 10** The self-mapping *f* is said to have the property *P* if $F({f}^{n})=F(f)$ for every $n\in \mathbb{N}$. A pair $(f,g)$ of self-mappings is said to have the property *Q* if $F(f)\cap F(g)=F({f}^{n})\cap F({g}^{n})$.

For further details on these properties, we refer to [20, 28].

Let $(X,d)$ be a metric space and $f:X\to X$ be a self-mapping. The set $O(x)=\{x,fx,\dots ,{f}^{n}x,\dots \}$ is called the orbit of *x* [29]. A mapping *f* is called orbitally continuous at *p* if ${lim}_{n\to \mathrm{\infty}}{f}^{n}x=p$ implies that ${lim}_{n\to \mathrm{\infty}}{f}^{n+1}x=fp$. A mapping *f* is orbitally continuous on *X* if *f* is orbitally continuous for all $x\in X$.

In this section we prove some periodic point results for self-mappings on complete metric spaces.

**Theorem 4**

*Let*

*X*

*be a nonempty set such that there exists a complete metric*

*d*

*on*

*X*.

*Suppose that*$f:X\to X$

*satisfies*

*for some* $\tau >0$ *and for all* *x* *in* *X* *such that* $d(fx,{f}^{2}x)>0$. *Then* *f* *has the property* *P* *provided that* *f* *is orbitally continuous on* *X*.

*Proof*First we show that $F(f)\ne \mathrm{\varnothing}$. Let ${x}_{0}\in X$. Define a sequence ${\{{x}_{n}\}}_{n\in \mathbb{N}}$ in

*X*, such that ${x}_{n+1}=f{x}_{n}$, for all $n\in \mathbb{N}\cup \{0\}$. Denote ${\gamma}_{n}=d({x}_{n},{x}_{n+1})$ for all $n\in \mathbb{N}\cup \{0\}$. If there exists ${n}_{0}\in \mathbb{N}\cup \{0\}$ for which ${x}_{{n}_{0}+1}={x}_{{n}_{0}}$, then $f{x}_{{n}_{0}}={x}_{{n}_{0}}$ and the proof is finished. Suppose that ${x}_{n+1}\ne {x}_{n}$ for all $n\in \mathbb{N}\cup \{0\}$. Using (9), we obtain

*X*is complete, which implies that there exists

*x*in

*X*such that ${lim}_{n\to \mathrm{\infty}}{f}^{n}{x}_{0}=x$. Since

*f*is orbitally continuous at

*x*, so $x={lim}_{n\to \mathrm{\infty}}{f}^{n}{x}_{0}=f({lim}_{n\to \mathrm{\infty}}{f}^{n-1}{x}_{0})=fx$. Hence

*f*has a fixed point and $F({f}^{n})=F(f)$ is true for $n=1$. Now assume $n>1$. Suppose on the contrary that $u\in F({f}^{n})$ but $u\notin F(f)$, then $d(u,fu)=\alpha >0$. Now consider

Thus $F(\alpha )\le {lim}_{n\to \mathrm{\infty}}F(d(u,fu))-n\tau =-\mathrm{\infty}$. Hence $F(\alpha )=-\mathrm{\infty}$. By (C2) $\alpha =0$, a contradiction. So $u\in F(f)$. □

**Theorem 5**

*Let*$(X,\u2aaf)$

*be a partially ordered set such that there exists a complete metric*

*d*

*on*

*X*

*and*

*f*,

*g*

*self*-

*mappings on*

*X*.

*Further assume that*

*f*,

*g*

*are weakly increasing and satisfy*

*for some* $\tau >0$, *for all* *x*, *y* *in* *X* *such that* $min\{d(fx,gy),d(x,y)\}>0$. *Then* *f* *and* *g* *have the property* *Q* *provided that* *X* *has the sequential limit comparison property*.

*Proof*By Theorem 3,

*f*and

*g*have a common fixed point. Suppose on the contrary that

As ${lim}_{n\to \mathrm{\infty}}F(d(u,gu))-n\tau =-\mathrm{\infty}$, so we have $F(\alpha )=-\mathrm{\infty}$. By (C2) $\alpha =0$, a contradiction. Hence $u\in F(g)\cap F(f)$. □

## Declarations

### Acknowledgements

The third author thanks for the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.

## Authors’ Affiliations

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