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# Generalized *α*-*ψ* contractive mappings in quasi-metric spaces and related fixed-point theorems

- Nurcan Bilgili
^{1}, - Erdal Karapınar
^{2}Email author and - Bessem Samet
^{3}

**2014**:36

https://doi.org/10.1186/1029-242X-2014-36

© Bilgili et al.; licensee Springer. 2014

**Received:**27 October 2013**Accepted:**26 December 2013**Published:**24 January 2014

## Abstract

In this paper, we characterize *α*-*ψ* contractive mappings in the setting of quasi-metric spaces and investigate the existence and uniqueness of a fixed point of such mappings. We notice that by using our result some fixed-point theorems in the context of *G*-metric space can be deduced.

**MSC:**46T99, 47H10, 54H25, 46J10.

## Keywords

- fixed point
*α*-*ψ*contractive mappings- quasi-metric spaces

## 1 Introduction and preliminaries

Let Ψ be the family of functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ satisfying the following conditions:

(${\psi}_{1}$) *ψ* is nondecreasing;

(${\psi}_{2}$) ${\sum}_{n=1}^{+\mathrm{\infty}}{\psi}^{n}(t)<\mathrm{\infty}$ for all $t>0$, where ${\psi}^{n}$ is the *n* th iterate of *ψ*.

These functions are known in the literature as (c)-comparison functions. One can easily deduce that if *ψ* is a (c)-comparison function, then $\psi (t)<t$ for any $t>0$.

**Definition 1** Let *X* be a non-empty and let $d:X\times X\to [0,\mathrm{\infty})$ be a function which satisfies:

(d1) $d(x,y)=0$ if and only if $x=y$,

(d2) $d(x,y)\le d(x,z)+d(z,y)$. Then *d* called a quasi-metric and the pair $(X,d)$ is called a quasi-metric space.

**Remark 2** Any metric space is a quasi-metric space, but the converse is not true in general.

Now, we give convergence and completeness on quasi-metric spaces.

**Definition 3**Let $(X,d)$ be a quasi-metric space, $\{{x}_{n}\}$ be a sequence in

*X*, and $x\in X$. The sequence $\{{x}_{n}\}$ converges to

*x*if and only if

**Definition 4** Let $(X,d)$ be a quasi-metric space and $\{{x}_{n}\}$ be a sequence in *X*. We say that $\{{x}_{n}\}$ is left-Cauchy if and only if for every $\epsilon >0$ there exists a positive integer $N=N(\epsilon )$ such that $d({x}_{n},{x}_{m})<\epsilon $ for all $n\ge m>N$.

**Definition 5** Let $(X,d)$ be a quasi-metric space and $\{{x}_{n}\}$ be a sequence in *X*. We say that $\{{x}_{n}\}$ is right-Cauchy if and only if for every $\epsilon >0$ there exists a positive integer $N=N(\epsilon )$ such that $d({x}_{n},{x}_{m})<\epsilon $ for all $m\ge n>N$.

**Definition 6** Let $(X,d)$ be a quasi-metric space and $\{{x}_{n}\}$ be a sequence in *X*. We say that $\{{x}_{n}\}$ is Cauchy if and only if for every $\epsilon >0$ there exists a positive integer $N=N(\epsilon )$ such that $d({x}_{n},{x}_{m})<\epsilon $ for all $m,n>N$.

**Remark 7** A sequence $\{{x}_{n}\}$ in a quasi-metric space is Cauchy if and only if it is left-Cauchy and right-Cauchy.

**Definition 8**Let $(X,d)$ be a quasi-metric space. We say that

- (1)
$(X,d)$ is left-complete if and only if each left-Cauchy sequence in

*X*is convergent. - (2)
$(X,d)$ is right-complete if and only if each right-Cauchy sequence in

*X*is convergent. - (3)
$(X,d)$ is complete if and only if each Cauchy sequence in

*X*is convergent.

**Definition 9**Let $(X,d)$ be a quasi-metric space and $T:X\to X$ be a given mapping. We say that

*T*is an

*α*-

*ψ*contractive mapping if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that

**Remark 10** We easily see that any contractive mapping, that is, a mapping satisfying the Banach contraction, is an *α*-*ψ* contractive mapping with $\alpha (x,y)=1$ for all $x,y\in X$ and $\psi (t)=kt$, $k\in (0,1)$.

**Definition 11**Let $T:X\to X$ and $\alpha :X\times X\to [0,\mathrm{\infty})$. We say that

*T*is

*α*admissible if for all $x,y\in X$ we have

## 2 Main results

We start this section by the following definition, which is a characterization of *α*-*ψ* contractive mappings [1] in the context of a quasi-metric space.

**Definition 12** (*cf.* [2])

*T*is an

*α*-

*ψ*contractive mapping if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$, we have

**Theorem 13**

*Let*$(X,d)$

*be a complete quasi*-

*metric space*.

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

*is a*

*α*-

*ψ*

*contractive mapping which satisfies*

- (i)
*T**is**α**admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha (T{x}_{0},{x}_{0})\ge 1$*and*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (iii)
*T**is continuous*.

*Then* *T* *has a fixed point*.

*Proof*By (ii), there exists ${x}_{0}\in X$ such that $\alpha (T{x}_{0},{x}_{0})\ge 1$ and $\alpha ({x}_{0},T{x}_{0})\ge 1$. Let us define a sequence $\{{x}_{n}\}$ in

*X*by ${x}_{n+1}=T{x}_{n}$ for all $n\in \mathbb{N}$. If ${x}_{{n}_{0}}={x}_{{n}_{0}+1}$ for some ${n}_{0}$, then it is evident that ${x}_{{n}_{0}}$ is a fixed point of

*T*. Consequently, throughout the proof, we suppose that ${x}_{n}\ne {x}_{n+1}$ for all $n\in \mathbb{N}$. Regarding the assumption (i), we derive

Letting $n\to \mathrm{\infty}$ in the above inequality, we derive ${\sum}_{p=n}^{\mathrm{\infty}}{\psi}^{n}(d({x}_{1},{x}_{0}))\to 0$. Hence, $d({x}_{n+k},{x}_{n})\to 0$ as $n\to \mathrm{\infty}$. Therefore, $\{{x}_{n}\}$ is a left-Cauchy sequence in $(X,d)$.

*T*, we obtain

Keeping (2.12) and (2.15) in mind together with the uniqueness of the limit, we conclude that $u=Tu$, that is, *u* is a fixed point of *T*. □

**Theorem 14**

*Let*$(X,d)$

*be a complete quasi*-

*metric space*.

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

*is an*

*α*-

*ψ*

*contractive mapping which satisfies*:

- (i)
*T**is**α**admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha (T{x}_{0},{x}_{0})\ge 1$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\alpha ({x}_{n+1},{x}_{n})\ge 1$*for all**n**and*${x}_{n}\to x\in X$*as*$n\to \mathrm{\infty}$,*then there exists a subsequence*$\{{x}_{n(k)}\}$*of*$\{{x}_{n}\}$*such that*$\alpha (x,{x}_{n(k)})\ge 1$*for all k*.

*Then* *T* *has a fixed point*.

*Proof*Following the lines of the proof of Theorem 13, we know that the sequence $\{{x}_{n}\}$ defined by ${x}_{n+1}=T{x}_{n}$, for all $n\ge 0$, converges for some $u\in X$. From (2.3) and condition (iii), there exists a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that $\alpha (u,{x}_{n(k)})\ge 1$ for all

*k*. Applying (2.1), for all

*k*, we get

Thus, we have $d(Tu,u)=0$, that is, $Tu=u$. □

**Definition 15** (*cf.* [3])

*T*is a generalized

*α*-

*ψ*contractive mapping of type

*A*if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$ and we have

**Definition 16**Let $(X,d)$ be a quasi-metric space and $T:X\to X$ be a given mapping. We say that

*T*is a generalized

*α*-

*ψ*contractive mapping of type

*B*if there exist two functions $\alpha :X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$ and we have

**Theorem 17**

*Let*$(X,d)$

*be a complete quasi*-

*metric space*.

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

*is a generalized*

*α*-

*ψ*

*contractive mapping of type*

*A*

*and satisfies*

- (i)
*T**is**α**admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha (T{x}_{0},{x}_{0})\ge 1$*and*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (iii)
*T**is continuous*.

*Then* *T* *has a fixed point*.

*Proof*By assumption (ii), there exists ${x}_{0}\in X$ such that $\alpha (T{x}_{0},{x}_{0})\ge 1$ and $\alpha ({x}_{0},T{x}_{0})\ge 1$. We construct a sequence $\{{x}_{n}\}$ in

*X*in the following way:

*T*. Hence, we assume that ${x}_{n}\ne {x}_{n+1}$ for all $n\in \mathbb{N}$. Due to assumption (i), we have

*ψ*is a nondecreasing function, (2.24) implies that

Therefore, $\{{x}_{n}\}$ is a left-Cauchy sequence in $(X,d)$.

*ψ*is nondecreasing function, the inequality (2.33) turns into

for all $n\ge 1$. We shall examine three cases.

a contradiction.

*ψ*is a nondecreasing function, (2.24) implies that

for all $n\ge 1$.

Therefore, by (2.39) and (2.48), we conclude that $\{{x}_{n}\}$ is a right-Cauchy sequence in $(X,d)$.

*T*, we obtain

It follows from (2.49) and (2.52) that $u=Tu$, that is, *u* is a fixed point of *T*. □

**Corollary 18**

*Let*$(X,d)$

*be a complete quasi*-

*metric space*.

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

*is a generalized*

*α*-

*ψ*

*contractive mapping of type*

*A*

*and satisfies*:

- (i)
*T**is**α**admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha (T{x}_{0},{x}_{0})\ge 1$*and*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (iii)
*T**is continuous*.

*Then* *T* *has a fixed point*.

*ψ*is nondecreasing and, hence,

where $M(x,y)$ and $N(x,y)$ are defined as in Definition 15 and Definition 16. The rest follows from Theorem 17.

In the following theorem we are able to remove the continuity condition for the *α*-*ψ* contractive mappings of type *B*.

**Theorem 19**

*Let*$(X,d)$

*be a complete quasi*-

*metric space*.

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

*is a generalized*

*α*-

*ψ*

*contractive mapping of type*

*B*

*which satisfies*:

- (i)
*T**is**α**admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\alpha (T{x}_{0},{x}_{0})\ge 1$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\alpha ({x}_{n+1},{x}_{n})\ge 1$*for all**n**and*${x}_{n}\to x\in X$*as*$n\to \mathrm{\infty}$,*then there exists a subsequence*$\{{x}_{n(k)}\}$*of*$\{{x}_{n}\}$*such that*$\alpha (x,{x}_{n(k)})\ge 1$*for all k*.

*Then* *T* *has a fixed point*.

*Proof*Following the lines in the proof of Theorem 17, we know that the sequence $\{{x}_{n}\}$ defined by ${x}_{n+1}=T{x}_{n}$ for all $n\ge 0$, converges for some $u\in X$. From (2.23) and condition (iii), there exists a subsequence $\{{x}_{n(k)}\}$ of $\{{x}_{n}\}$ such that $\alpha (u,{x}_{n(k)})\ge 1$ for all

*k*. Applying (2.20), for all

*k*, we get

*k*large enough, we have $N(u,{x}_{n(k)})>0$, which implies that $\psi (N(u,{x}_{n(k)}))<N(u,{x}_{n(k)})$. Then, from (2.53), we have

which is a contradiction. Therefore, we find $d(Tu,u)=0$, that is, $Tu=u$. □

## 3 Consequences: fixed-point result on *G*-metric spaces

In this section, we note that some existing fixed-point results in the context of *G*-metric spaces are consequences of our main theorems. For the sake of completeness, we recollect some basic definitions and crucial results on the topic in the literature. For more details, see *e.g.* [4–6].

**Definition 20** Let *X* be a non-empty set, $G:X\times X\times X\to {\mathbb{R}}^{+}$ be a function satisfying the following properties:

(G1) $G(x,y,z)=0$ if $x=y=z$,

(G2) $0<G(x,x,y)$ for all $x,y\in X$ with $x\ne y$,

(G3) $G(x,x,y)\le G(x,y,z)$ for all $x,y,z\in X$ with $y\ne z$,

(G4) $G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots $ (symmetry in all three variables),

(G5) $G(x,y,z)\le G(x,a,a)+G(a,y,z)$ (rectangle inequality) for all $x,y,z,a\in X$.

Then the function *G* is called a generalized metric, or, more specifically, a *G*-metric on *X*, and the pair $(X,G)$ is called a *G*-metric space.

*G*-metric on

*X*induces a metric ${d}_{G}$ on

*X*defined by

For a better understanding of the subject we give the following examples of *G*-metrics.

**Example 21**Let $(X,d)$ be a metric space. The function $G:X\times X\times X\to [0,+\mathrm{\infty})$, defined by

for all $x,y,z\in X$, is a *G*-metric on *X*.

**Example 22**Let $X=[0,\mathrm{\infty})$. The function $G:X\times X\times X\to [0,+\mathrm{\infty})$, defined by

for all $x,y,z\in X$, is a *G*-metric on *X*.

**Definition 23**Let $(X,G)$ be a

*G*-metric space, and let $\{{x}_{n}\}$ be a sequence of points of

*X*. We say that $\{{x}_{n}\}$ is

*G*-convergent to $x\in X$ if

that is, for any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G(x,{x}_{n},{x}_{m})<\epsilon $, for all $n,m\ge N$. We call *x* the limit of the sequence and write ${x}_{n}\to x$ or ${lim}_{n\to +\mathrm{\infty}}{x}_{n}=x$.

**Proposition 24**

*Let*$(X,G)$

*be a*

*G*-

*metric space*.

*The following are equivalent*:

- (1)
$\{{x}_{n}\}$

*is**G*-*convergent to**x*, - (2)
$G({x}_{n},{x}_{n},x)\to 0$

*as*$n\to +\mathrm{\infty}$, - (3)
$G({x}_{n},x,x)\to 0$

*as*$n\to +\mathrm{\infty}$, - (4)
$G({x}_{n},{x}_{m},x)\to 0$

*as*$n,m\to +\mathrm{\infty}$.

**Definition 25** Let $(X,G)$ be a *G*-metric space. A sequence $\{{x}_{n}\}$ is called a *G*-Cauchy sequence if, for any $\epsilon >0$, there exists $N\in \mathbb{N}$ such that $G({x}_{n},{x}_{m},{x}_{l})<\epsilon $ for all $m,n,l\ge N$, that is, $G({x}_{n},{x}_{m},{x}_{l})\to 0$ as $n,m,l\to +\mathrm{\infty}$.

**Proposition 26**

*Let*$(X,G)$

*be a*

*G*-

*metric space*.

*Then the following are equivalent*:

- (1)
*the sequence*$\{{x}_{n}\}$*is**G*-*Cauchy*, - (2)
*for any*$\epsilon >0$,*there exists*$N\in \mathbb{N}$*such that*$G({x}_{n},{x}_{m},{x}_{m})<\epsilon $,*for all*$m,n\ge N$.

**Definition 27** A *G*-metric space $(X,G)$ is called *G*-complete if every *G*-Cauchy sequence is *G*-convergent in $(X,G)$.

For more details of *G*-metric space, we refer *e.g.* to [7–9].

**Theorem 28**

*Let*$(X,G)$

*be a*

*G*-

*metric space*.

*Let*$d:X\times X\to [0,\mathrm{\infty})$

*be the function defined by*$d(x,y)=G(x,y,y)$.

*Then*

- (1)
$(X,d)$

*is a quasi*-*metric space*; - (2)
$\{{x}_{n}\}\subset X$

*is**G*-*convergent to*$x\in X$*if and only if*$\{{x}_{n}\}$*is convergent to**x**in*$(X,d)$; - (3)
$\{{x}_{n}\}\subset X$

*is**G*-*Cauchy if and only if*$\{{x}_{n}\}$*is Cauchy in*$(X,d)$; - (4)
$(X,G)$

*is**G*-*complete if and only if*$(X,d)$*is complete*.

is a metric on *X*.

As an immediate consequence of the definition above and Theorem 28, the following theorem is obtained.

**Theorem 29**

*Let*$(X,G)$

*be a*

*G*-

*metric space*.

*Let*$d:X\times X\to [0,\mathrm{\infty})$

*be the function defined by*$d(x,y)=G(x,y,y)$.

*Then*

- (1)
$(X,d)$

*is a quasi*-*metric space*; - (2)
$\{{x}_{n}\}\subset X$

*is**G*-*convergent to*$x\in X$*if and only if*$\{{x}_{n}\}$*is convergent to**x**in*$(X,\delta )$; - (3)
$\{{x}_{n}\}\subset X$

*is**G*-*Cauchy if and only if*$\{{x}_{n}\}$*is Cauchy in*$(X,\delta )$; - (4)
$(X,G)$

*is**G*-*complete if and only if*$(X,\delta )$*is complete*.

Now, we state the characterization of Definition 9 and Definition 11 in the context of *G*-metric space.

**Definition 30** (See *e.g.* [10, 11])

*G*-metric space and $T:X\to X$ be a given mapping. We say that

*T*is a

*β*-

*ψ*contractive mapping of type I if there exist two functions $\beta :X\times X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$, we have

**Definition 31** (See *e.g.* [10, 11])

*T*is

*β*admissible if for all $x,y\in X$ we have

**Lemma 32** *Let* $T:X\to X$ *where* *X* *is non*-*empty set*. *It is clear that the self*-*mapping* *T* *is* *β* *admissible if and only if* *T* *is* *α* *admissible*.

*Proof* It is sufficient to let $\alpha (x,y)=\beta (x,y,y)$. □

**Theorem 33**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space*.

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

*is a*

*β*-

*ψ*

*contractive mapping which satisfies*:

- (i)
*T**is**β**admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\beta (T{x}_{0},{x}_{0},{x}_{0})\ge 1$*and*$\beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1$; - (iii)
*T**is continuous*.

*Then* *T* *has a fixed point*.

*Proof*Consider the quasi-metric $d(x,y)=G(x,y,y)$ for all $x,y\in X$. Due to Lemma 32 and (3.2), we have

Then the result follows from Theorem 13. □

**Definition 34**Let $(X,G)$ be a

*G*-metric space and $T:X\to X$ be a given mapping. We say that

*T*is a generalized

*β*-

*ψ*contractive mapping of type

*A*if there exist two functions $\beta :X\times X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$ we have

**Definition 35**Let $(X,G)$ be a

*G*-metric space and $T:X\to X$ be a given mapping. We say that

*T*is a generalized

*β*-

*ψ*contractive mapping of type

*B*if there exist two functions $\beta :X\times X\times X\to [0,\mathrm{\infty})$ and $\psi \in \mathrm{\Psi}$ such that for all $x,y\in X$ we have

**Remark 36** It is simple to see that every *β*-*ψ* contractive mapping is a generalized *β*-*ψ* contractive mapping of type *A*.

Similarly, every *β*-*ψ* contractive mapping is a generalized *β*-*ψ* contractive mapping of type *B*.

**Theorem 37**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space*.

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

*is a generalized*

*β*-

*ψ*

*contractive mapping of type*

*A*

*and satisfies*:

- (i)
*T**is**β**admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\beta (T{x}_{0},{x}_{0},{x}_{0})\ge 1$*and*$\beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1$; - (iii)
*T**is continuous*.

*Then* *T* *has a fixed point*.

*Proof*Consider the quasi-metric $d(x,y)=G(x,y,y)$ for all $x,y\in X$. From Lemma 32 together with (3.5) and (3.6), we deduce that

Then the result follows from Theorem 17. □

**Theorem 38**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space*.

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

*is a*

*β*-

*ψ*

*contractive mapping which satisfies*:

- (i)
*T**is**β**admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\beta (T{x}_{0},{x}_{0},{x}_{0})\ge 1$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\beta ({x}_{n+1},{x}_{n},{x}_{n})\ge 1$*for all**n**and*${x}_{n}\to x\in X$*as*$n\to \mathrm{\infty}$,*then there exists a subsequence*$\{{x}_{n(k)}\}$*of*$\{{x}_{n}\}$*such that*$\beta (x,{x}_{n(k)},{x}_{n(k)})\ge 1$*for all**k*.

*Then* *T* *has a fixed point*.

*Proof*Consider the quasi-metric $d(x,y)=G(x,y,y)$ for all $x,y\in X$. By Lemma 32 and (3.2), we find that

Then the result follows from Theorem 14. □

**Theorem 39**

*Let*$(X,G)$

*be a complete*

*G*-

*metric space*.

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

*is a generalized*

*β*-

*ψ*

*contractive mapping of type*

*B*

*which satisfies*:

- (i)
*T**is**β**admissible*; - (ii)
*there exists*${x}_{0}\in X$*such that*$\beta (T{x}_{0},{x}_{0},{x}_{0})\ge 1$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$\beta ({x}_{n+1},{x}_{n},{x}_{n})\ge 1$*for all**n**and*${x}_{n}\to x\in X$*as*$n\to \mathrm{\infty}$,*then there exists a subsequence*$\{{x}_{n(k)}\}$*of*$\{{x}_{n}\}$*such that*$\beta (x,{x}_{n(k)},{x}_{n(k)})\ge 1$*for all**k*.

*Then* *T* *has a fixed point*.

*Proof*Consider the quasi-metric $d(x,y)=G(x,y,y)$ for all $x,y\in X$. Regarding Lemma 32, (3.7), and (3.8), we derive

Then the result follows from Theorem 19. □

## Declarations

### Acknowledgements

The authors thank the Visiting Professor Programming at King Saud University for funding this work. The authors thank the anonymous referees for their remarkable comments, suggestion, and ideas that helped to improve this paper.

## Authors’ Affiliations

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