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# Common fixed point results for *α*-*ψ*-contractions on a metric space endowed with graph

- Nawab Hussain
^{1}Email author, - Muhammad Arshad
^{2}, - Abdullah Shoaib
^{2}and - Fahimuddin
^{2}

**2014**:136

https://doi.org/10.1186/1029-242X-2014-136

© Hussain et al.; licensee Springer. 2014

**Received:**26 December 2013**Accepted:**14 March 2014**Published:**31 March 2014

## Abstract

Abdeljawad (Fixed Point Theory Appl., 2013:19) introduced the concept of *α*-admissible for a pair of mappings. More recently Salimi *et al.* [Fixed Point Theory Appl., 2013:151] modified the notion of *α*-*ψ*-contractive mappings. In this paper we introduce the concept of an *α*-admissible map with respect to *η* and modify the *α*-*ψ*-contractive condition for a pair of mappings and establish common fixed point results for two, three, and four mappings in a closed ball in complete dislocated metric spaces. As an application, we derive some new common fixed point theorems for *ψ*-graphic contractions defined on dislocated metric space endowed with a graph as well as preordered dislocated metric space. Some comparative examples are constructed which illustrate the superiority of our results to the existing ones in the literature.

**MSC:** 46S40, 47H10, 54H25.

## Keywords

- common fixed point
- complete dislocated metric space
*α*-*ψ*-contractive mappings*ψ*-graphic contractions- closed ball

## 1 Introduction and preliminaries

Fixed point results of mappings satisfying certain contractive condition on the entire domain has been at the center of rigorous research activities, for example, see [1–33]. From application point of view the situation is not yet completely satisfactory because it frequently happens that a mapping *T* is a contraction not on the entire space *X* but merely on a subset *Y* of *X*. Recently Arshad *et al.* [8] proved a result concerning the existence of fixed points of a mapping satisfying a contractive condition on closed ball in a complete dislocated metric space (see also [9, 14, 15, 25, 33]). The notion of dislocated topologies has useful applications in the context of logic programming semantics (see [5, 17, 29]).

The existence of fixed points of *α*-*ψ*-contractive and *α*-admissible mappings in complete metric spaces has been studied by several researchers (see [18–20] and references therein). In this paper we discuss common fixed point results for *α*-*ψ*-contractive type mappings in a closed ball in complete dislocated metric space. Our results improve several well known recent conventional results in [2, 8, 31]. We also derive some new common fixed point theorems for *ψ*-graphic contractions as well as ordered contractions on preordered metric space. We give examples which show how these results can be used when the corresponding results cannot.

Consistent with [2, 7, 8, 17, 31], the following definitions and results will be needed in the sequel.

**Definition 1.1** [17]

*X*be a non-empty set and let ${d}_{l}:X\times X\to [0,\mathrm{\infty})$ be a function, called a dislocated metric (or simply ${d}_{l}$-metric) if the following conditions hold for any $x,y,z\in X$:

- (i)
if ${d}_{l}(x,y)=0$, then $x=y$;

- (ii)
${d}_{l}(x,y)={d}_{l}(y,x)$;

- (iii)
${d}_{l}(x,y)\le {d}_{l}(x,z)+{d}_{l}(z,y)$.

The pair $(X,{d}_{l})$ is then called a dislocated metric space. It is clear that if ${d}_{l}(x,y)=0$, then from (i), $x=y$. But if $x=y$, ${d}_{l}(x,y)$ may not be 0.

**Definition 1.2** [17]

A sequence $\{{x}_{n}\}$ in a ${d}_{l}$-metric space $(X,{d}_{l})$ is called a Cauchy sequence if given $\epsilon >0$, there corresponds ${n}_{0}\in N$ such that for all $n,m\ge {n}_{0}$ we have ${d}_{l}({x}_{m},{x}_{n})<\epsilon $.

**Definition 1.3** [17]

A sequence $\{{x}_{n}\}$ in ${d}_{l}$-metric space converges with respect to ${d}_{l}$ if there exists $x\in X$ such that ${d}_{l}({x}_{n},x)\to 0$ as $n\to \mathrm{\infty}$. In this case, *x* is called a limit of $\{{x}_{n}\}$ and we write ${x}_{n}\to x$.

**Definition 1.4** [17]

A ${d}_{l}$-metric space $(X,{d}_{l})$ is called complete if every Cauchy sequence in *X* converges to a point in *X*.

**Definition 1.5** Let *X* be a non-empty set and $T,f:X\to X$. A point $y\in X$ is called point of coincidence of *T* and *f* if there exists a point $x\in X$ such that $y=Tx=fx$, here *x* is called coincidence point of *T* and *f*. The mappings *T*, *f* are said to be weakly compatible if they commute at their coincidence point (*i.e.*, $Tfx=fTx$ whenever $Tx=fx$).

We require the following lemmas for subsequent use.

**Lemma 1.6** [8]

*Let* *X* *be a non*-*empty set and* $f:X\to X$ *be a function*. *Then there exists* $E\subset X$ *such that* $fE=fX$ *and* $f:E\to X$ *is one*-*to*-*one*.

**Lemma 1.7** [7]

*Let* *X* *be a non*-*empty set and the mappings* $S,T,f:X\to X$ *have a unique point of coincidence* *v* *in* *X*. *If* $(S,f)$ *and* $(T,f)$ *are weakly compatible*, *then* *S*, *T*, *f* *have a unique common fixed point*.

Let Ψ denote the family of all nondecreasing functions $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ such that ${\sum}_{n=1}^{+\mathrm{\infty}}{\psi}^{n}(t)<+\mathrm{\infty}$ for all $t>0$, where ${\psi}^{n}$ is the *n* th iterate of *ψ*.

**Lemma 1.8** [31]

*If* $\psi \in \mathrm{\Psi}$, *then* $\psi (t)<t$ *for all* $t>0$.

**Definition 1.9** [2]

Let $S,T:X\to X$ and $\alpha :X\times X\to [0,+\mathrm{\infty})$. We say that the pair $(S,T)$ is *α*-admissible if $x,y\in X$ such that $\alpha (x,y)\ge 1$, then we have $\alpha (Sx,Ty)\ge 1$ and $\alpha (Tx,Sy)\ge 1$.

**Definition 1.10** [31]

Let $T:X\to X$ and $\alpha ,\eta :X\times X\to [0,+\mathrm{\infty})$ two functions. We say that *T* is *α*-admissible mapping with respect to *η* if $x,y\in X$ such that $\alpha (x,y)\ge \eta (x,y)$, then we have $\alpha (Tx,Ty)\ge \eta (Tx,Ty)$. Note that if we take $\eta (x,y)=1$, then *T* is called an *α*-admissible mapping [32].

## 2 Common fixed point results in dislocated metric space

We first extend the concept of *α*-*η*-admissibility for the pair of mappings.

**Definition 2.1** Let $S,T:X\to X$ and $\alpha ,\eta :X\times X\to [0,+\mathrm{\infty})$ two functions. We say that the pair $(S,T)$ is *α*-admissible with respect to *η* if $x,y\in X$ such that $\alpha (x,y)\ge \eta (x,y)$, then we have $\alpha (Sx,Ty)\ge \eta (Sx,Ty)$ and $\alpha (Tx,Sy)\ge \eta (Tx,Sy)$. Also, if we take $\eta (x,y)=1$, then the pair $(S,T)$ is called *α*-admissible, if we take, $\alpha (x,y)=1$, then we say that the pair $(S,T)$ is *η*-subadmissible mapping. If we take $\eta (x,y)=1$, then we obtain Definition 1 of Abdeljawad [2]. Also, if we take $S=T$, we obtain Definition 1.10.

**Theorem 2.2**

*Let*$(X,{d}_{l})$

*be a complete dislocated metric space and*$S,T:X\to X$

*be two mappings*.

*Suppose there exist two functions*, $\alpha ,\eta :X\times X\to [0,+\mathrm{\infty})$

*such that the pair*$(S,T)$

*is*

*α*-

*admissible with respect to*

*η*.

*For*$r>0$, ${x}_{0}\in \overline{B({x}_{0},r)}$,

*and*$\psi \in \mathrm{\Psi}$,

*assume that*

*and*

*Suppose that the following assertions hold*:

- (i)
$\alpha ({x}_{0},S{x}_{0})\ge \eta ({x}_{0},S{x}_{0})$;

- (ii)
*for any sequence*$\{{x}_{n}\}$*in*$\overline{B({x}_{0},r)}$*such that*$\alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1})$*for all*$n\in N\cup \{0\}$*and*${x}_{n}\to u\in \overline{B({x}_{0},r)}$*as*$n\to +\mathrm{\infty}$*then*$\alpha ({x}_{n},u)\ge \eta ({x}_{n},u)$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then there exists a point* ${x}^{\ast}$ *in* $\overline{B({x}_{0},r)}$ *such that* ${x}^{\ast}=S{x}^{\ast}=T{x}^{\ast}$.

*Proof*Let ${x}_{1}$ in

*X*be such that ${x}_{1}=S{x}_{0}$ and ${x}_{2}=T{x}_{1}$. Continuing this process, we construct a sequence ${x}_{n}$ of points in

*X*such that

*α*-admissible with respect to

*η*, we have, $\alpha (S{x}_{0},T{x}_{1})\ge \eta (S{x}_{0},T{x}_{1})$ from which we deduce that $\alpha ({x}_{1},{x}_{2})\ge \eta ({x}_{1},{x}_{2})$ which also implies that $\alpha (T{x}_{1},S{x}_{2})\ge \eta (T{x}_{1},S{x}_{2})$. Continuing in this way we obtain $\alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1})$ for all $n\in \mathbb{N}\cup \{0\}$. First, we show that ${x}_{n}\in \overline{B({x}_{0},r)}$ for all $n\in \mathbb{N}$. Using inequality (2), we have

we obtain ${d}_{l}(T{x}^{\ast},{x}^{\ast})=0$, that is, $T{x}^{\ast}={x}^{\ast}$. Hence *S* and *T* have a common fixed point in $\overline{B({x}_{0},r)}$. □

If $\eta (x,y)=1$ for all $x,y\in X$ in Theorem 2.2, we obtain the following result.

**Corollary 2.3**

*Let*$(X,{d}_{l})$

*be a complete dislocated metric space and*$S,T:X\to X$, $r>0$

*and*${x}_{0}$

*be an arbitrary point in*$\overline{B({x}_{0},r)}$.

*Suppose there exists*$\alpha :X\times X\to [0,+\mathrm{\infty})$

*such that the pair*$(S,T)$

*is*

*α*-

*admissible*.

*For*$\psi \in \mathrm{\Psi}$,

*assume that*

*and*

*Suppose that the following assertions hold*:

- (i)
$\alpha ({x}_{0},S{x}_{0})\ge 1$;

- (ii)
*for any sequence*$\{{x}_{n}\}$*in*$\overline{B({x}_{0},r)}$*such that*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to u\in \overline{B({x}_{0},r)}$*as*$n\to +\mathrm{\infty}$*then*$\alpha ({x}_{n},u)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then there exists a point* ${x}^{\ast}$ *in* $\overline{B({x}_{0},r)}$ *such that* ${x}^{\ast}=S{x}^{\ast}=T{x}^{\ast}$.

If $\alpha (x,y)=1$ for all $x,y\in X$ in Theorem 2.2, we obtain following result.

**Corollary 2.4**

*Let*$(X,{d}_{l})$

*be a complete dislocated metric space and*$S,T:X\to X$

*be two mappings*.

*Suppose there exists*$\eta :X\times X\to [0,+\mathrm{\infty})$

*such that the pair*$(S,T)$

*is*

*η*-

*subadmissible*.

*For*$\psi \in \mathrm{\Psi}$

*and*${x}_{0}\in \overline{B({x}_{0},r)}$,

*assume that*

*and*

*Suppose that the following assertions hold*:

- (i)
$\eta ({x}_{0},S{x}_{0})\le 1$;

- (ii)
*for any sequence*$\{{x}_{n}\}$*in*$\overline{B({x}_{0},r)}$*such that*$\eta ({x}_{n},{x}_{n+1})\le 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to u\in \overline{B({x}_{0},r)}$*as*$n\to +\mathrm{\infty}$*then*$\eta ({x}_{n},u)\le 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then there exists a point* ${x}^{\ast}$ *in* $\overline{B({x}_{0},r)}$ *such that* ${x}^{\ast}=S{x}^{\ast}=T{x}^{\ast}$.

**Corollary 2.5** (Theorem 2.2 of [32])

*Let*$(X,d)$

*be a complete metric space and*$S:X\to X$

*be an*

*α*-

*admissible mapping*.

*Assume that for*$\psi \in \mathrm{\Psi}$,

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

*Also*,

*suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*$\alpha ({x}_{0},S{x}_{0})\ge 1$; - (ii)
*for any sequence*$\{{x}_{n}\}$*in**X**with*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then* *S* *has a fixed point*.

**Theorem 2.6** *On adding the condition ‘if* ${x}^{\ast}$ *is any common fixed point in* $\overline{B({x}_{0},r)}$ *of* *S* *and* *T*, *x* *be any fixed point of* *S* *or* *T* *in* $\overline{B({x}_{0},r)}$, *then* $\alpha (x,{x}^{\ast})\ge \eta (x,{x}^{\ast})$*’ to the hypotheses of Theorem * 2.2, *S* *and* *T* *have a unique common fixed point* ${x}^{\ast}$ *and* ${d}_{l}({x}^{\ast},{x}^{\ast})=0$.

*Proof*Assume that ${y}^{\ast}$ be another fixed point of

*T*in $\overline{B({x}_{0},r)}$, then, by assumption, $\alpha ({x}^{\ast},{y}^{\ast})\ge \eta ({x}^{\ast},{y}^{\ast})$,

*T*has no fixed point other than ${x}^{\ast}$. Similarly,

*S*has no fixed point other than ${x}^{\ast}$. Now, $\alpha ({x}^{\ast},{x}^{\ast})\ge \eta ({x}^{\ast},{x}^{\ast})$, then

□

**Example 2.7**Let $X={Q}^{+}\cup \{0\}$ and ${d}_{l}:X\times X\to X$ be defined by ${d}_{l}(x,y)=x+y$. Then $(X,{d}_{l})$ is complete dislocated metric space (see [8]). Let $S,T:X\to X$ be defined by

*X*. Also, if $x,y\in \overline{B({x}_{0},r)}$, then

Therefore, all the conditions of Corollary 2.3 are satisfied and *S* and *T* have a common fixed point 0.

Now we apply our Theorem 2.6 to obtain unique common fixed point of three mappings on a closed ball in complete dislocated metric space.

**Theorem 2.8**

*Let*$(X,{d}_{l})$

*be a dislocated metric space*, $S,T,f:X\to X$

*such that*$SX\cup TX\subset fX$, $r>0$

*and*${x}_{0}\in \overline{B({x}_{0},r)}$.

*Suppose there exist two functions*, $\alpha ,\eta :X\times X\to [0,+\mathrm{\infty})$

*α*-

*admissible with respect to*

*η*

*and*$\psi \in \mathrm{\Psi}$

*such that*

*and*

*Suppose that*

- (i)
*the pair*$(S,T)$*and**f**are**α*-*admissible with respect to**η*; - (ii)
$\alpha (f{x}_{0},S{x}_{0})\ge \eta (f{x}_{0},S{x}_{0})$;

- (iii)
*if*$\{{x}_{n}\}$*is a sequence in*$\overline{B(f{x}_{0},r)}$*such that*$\alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1})$*for all**n**and*${x}_{n}\to u\in \overline{B(f{x}_{0},r)}$*as*$n\to +\mathrm{\infty}$*then*$\alpha ({x}_{n},u)\ge \eta ({x}_{n},u)$*for all*$n\in \mathbb{N}\cup \{0\}$; - (iv)
*if**fx**is any point in*$\overline{B(f{x}_{0},r)}$*such that*$Sx=Tx=fx$*and**fy**be any point in*$\overline{B(f{x}_{0},r)}$*such that*$Sy=fy$*or*$Ty=fy$,*then*$\alpha (fx,fy)\ge \eta (fx,fy)$; - (v)
*fX**is complete subspace of**X**and*$(S,f)$*and*$(T,f)$*are weakly compatible*.

*Then* *S*, *T*, *and* *f* *have a unique common fixed point* *fz* *in* $\overline{B(f{x}_{0},r)}$. *Moreover*, ${d}_{l}(fz,fz)=0$.

*Proof*By Lemma 1.6, there exists $E\subset X$ such that $fE=fX$ and $f:E\to X$ is one-to-one. Now since $SX\cup TX\subset fX$, we define two mappings $g,h:fE\to fE$ by $g(fx)=Sx$ and $h(fx)=Tx$, respectively. Since

*f*is one-to-one on

*E*, then

*g*,

*h*are well defined. Now $f{x}_{0}\in \overline{B(f{x}_{0},r)}\subseteq fX$. Then $f{x}_{0}\in fX$. Let ${y}_{0}=f{x}_{0}$, choose a point ${y}_{1}$ in

*fX*such that ${y}_{1}=g({y}_{0})$ and let ${y}_{2}=h({y}_{1})$. Continuing this process and having chosen ${y}_{n}$ in

*fX*such that

*f*is

*α*-admissible then $\alpha (x,y)\ge \eta (x,y)$ implies

*α*-admissible then $\alpha (x,y)\ge \eta (x,y)$ implies

*α*-admissible. As $\alpha ({y}_{0},{y}_{1})\ge \eta ({y}_{0},{y}_{1})\u27f9\alpha (g{y}_{0},h{y}_{1})\ge \eta (g{y}_{0},h{y}_{1})\u27f9\alpha (h{y}_{1},g{y}_{2})\ge \eta (h{y}_{1},g{y}_{2})$. Continuing this process, we have $\alpha ({y}_{n},{y}_{n+1})\ge \eta ({y}_{n},{y}_{n+1})$. Following similar arguments to those of Theorem 2.2, ${y}_{n}\in \overline{B(f{x}_{0},r)}$. Also by inequality (10).

As *fX* is a complete space, all conditions of Theorem 2.6 are satisfied, we deduce that there exists a unique common fixed point $fz\in \overline{B(f{x}_{0},r)}$ of *g* and *h*. Now $fz=g(fz)=h(fz)$ or $fz=Sz=Tz=fz$. Thus *fz* is the point of coincidence of *S*, *T* and *f*. Let $v\in \overline{B(f{x}_{0},r)}$ be another point of coincidence of *f*, *S* and *T* then there exists $u\in \overline{B(f{x}_{0},r)}$ such that $v=fu=Su=Tu$, which implies that $fu=g(fu)=h(fu)$. A contradiction as $fz\in \overline{B(f{x}_{0},r)}$ is a unique common fixed point of *g* and *h*. Hence $v=fz$. Thus *S*, *T* and *f* have a unique point of coincidence $fz\in \overline{B(f{x}_{0},r)}$. Now since $(S,f)$ and $(T,f)$ are weakly compatible, by Lemma 1.7 *fz* is a unique common fixed point of *S*, *T*, and *f*. □

Similarly, we can apply our Theorem 2.6 to obtain unique common fixed point and point of coincidence of four mappings in complete dislocated metric space. One can easily obtain conclusion by using the technique given in the proof of Theorem 2.8 [8].

**Theorem 2.9**

*Let*$(X,{d}_{l})$

*be a dislocated metric space and*

*S*,

*T*,

*g*

*and*

*f*

*be self*-

*mappings on*

*X*

*such that*$SX,TX\subset fX=gX$, $r>0$

*and*${x}_{0}\in \overline{B({x}_{0},r)}$.

*Suppose there exist two functions*$\alpha ,\eta :X\times X\to [0,+\mathrm{\infty})$

*is*

*α*-

*admissible with respect to*

*η*

*and*$\psi \in \mathrm{\Psi}$

*such that*

*and*

*Suppose that*

- (i)
*the pairs*$(S,T)$*and*$(f,g)$*are**α*-*admissible with respect to**η*; - (ii)
$\alpha (f{x}_{0},S{x}_{0})\ge \eta (f{x}_{0},S{x}_{0})$;

- (iii)
*if*$\{{x}_{n}\}$*is a sequence in*$\overline{B(f{x}_{0},r)}$*such that*$\alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1})$*for all**n**and*${x}_{n}\to u\in \overline{B(f{x}_{0},r)}$*as*$n\to +\mathrm{\infty}$*then*$\alpha ({x}_{n},u)\ge \eta ({x}_{n},u)$*for all**n*; - (iv)
*if*$fx=gx$*is any point in*$\overline{B(f{x}_{0},r)}$*such that*$Sx=Tx=fx$*and*$fy=gy$*be any point in*$\overline{B(f{x}_{0},r)}$*such that*$Sy=fy$*or*$Ty=fy$,*then*$\alpha (fx,fy)\ge \eta (fx,fy)$; - (v)
*fX**is complete subspace of**Xand*$(S,f)$*and*$(T,g)$*are weakly compatible*.

*Then* *S*, *T*, *f*, *and* *g* *have a unique common fixed point* *fz* *in* $\overline{B(f{x}_{0},r)}$.

A partial metric version of Theorem 2.2 is given below.

**Theorem 2.10**

*Let*$(X,p)$

*be a complete partial metric space*, $S,T:X\to X$

*be two maps*, $r>0$

*and*${x}_{0}\in \overline{B({x}_{0},r)}$.

*Suppose there exist two functions*, $\alpha ,\eta :X\times X\to [0,+\mathrm{\infty})$

*such that*$(S,T)$

*be*

*α*-

*admissible with respect to*

*η*

*and*$\psi \in \mathrm{\Psi}$.

*Assume that*

*and*

*Suppose that the following assertions hold*:

- (i)
$\alpha ({x}_{0},S{x}_{0})\ge \eta ({x}_{0},S{x}_{0})$;

- (ii)
*for any sequence*$\{{x}_{n}\}$*in*$\overline{B({x}_{0},r)}$*such that*$\alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1})$*for all*$n\in N\cup \{0\}$*and*${x}_{n}\to u\in \overline{B({x}_{0},r)}$*as*$n\to +\mathrm{\infty}$*then*$\alpha ({x}_{n},u)\ge \eta ({x}_{n},u)$*for all*$n\in \mathbb{N}\cup \{0\}$.

*Then there exists a point* ${x}^{\ast}$ *in* $\overline{B({x}_{0},r)}$ *such that* ${x}^{\ast}=S{x}^{\ast}=T{x}^{\ast}$.

## 3 Fixed point results for graphic contractions in dislocated metric spaces

Consistent with Jachymski [24], let $(X,{d}_{l})$ be a dislocated metric space and Δ denotes the diagonal of the Cartesian product $X\times X$. Consider a directed graph *G* such that the set $V(G)$ of its vertices coincides with *X*, and the set $E(G)$ of its edges contains all loops, *i.e.*, $E(G)\supseteq \mathrm{\Delta}$. We assume *G* has no parallel edges, so we can identify *G* with the pair $(V(G),E(G))$. Moreover, we may treat *G* as a weighted graph (see [24]) by assigning to each edge the distance between its vertices. If *x* and *y* are vertices in a graph *G*, then a path in *G* from *x* to *y* of length *m* ($m\in \mathbb{N}$) is a sequence ${\{{x}_{i}\}}_{i=0}^{m}$ of $m+1$ vertices such that ${x}_{0}=x$, ${x}_{m}=y$ and $({x}_{n-1},{x}_{n})\in E(G)$ for $i=1,\dots ,m$. A graph *G* is connected if there is a path between any two vertices. *G* is weakly connected if $\tilde{G}$ is connected (see for details [1, 11, 21, 24]).

**Definition 3.1** [24]

*G*-contraction or simply

*G*-contraction if

*T*preserves the edges of

*G*,

*i.e.*,

*T*decreases the weights of the edges of

*G*in the following way:

Now we extend the concept of *G*-contraction for the pair of maps as follows.

**Definition 3.2**Let $(X,{d}_{l})$ be a dislocated metric space endowed with a graph

*G*and $S,T:X\to X$ be self-mappings. Assume that for $r>0$, ${x}_{0}\in \overline{B({x}_{0},r)}$ and $\psi \in \mathrm{\Psi}$ following conditions hold:

Then the mappings $(S,T)$ are called a *ψ*-graphic contractive mappings. If $\psi (t)=kt$ for some $k\in [0,1)$, then we say $(S,T)$ are *G*-contractive mappings.

**Theorem 3.3**

*Let*$(X,{d}_{l})$

*be a complete dislocated metric space endowed with a graph*

*G*

*and*$S,T:X\to X$

*be*

*ψ*-

*graphic contractive mappings and*${x}_{0}\in \overline{B({x}_{0},r)}$.

*Suppose that the following assertions hold*:

- (i)
$({x}_{0},S{x}_{0})\in E(G)$

*and*${\sum}_{i=0}^{j}{\psi}^{i}({d}_{l}({x}_{0},S{x}_{0}))\le r$*for all*$j\in \mathbb{N}$; - (ii)
*if*$\{{x}_{n}\}$*is a sequence in*$\overline{B({x}_{0},r)}$*such that*$({x}_{n},{x}_{n+1})\in E(G)$*for all*$n\in \mathbb{N}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*then*$({x}_{n},x)\in E(G)$*for all*$n\in \mathbb{N}$.

*Then* *S* *and* *T* *have a common fixed point*.

*Proof* Define, $\alpha :{X}^{2}\to (-\mathrm{\infty},+\mathrm{\infty})$ by $\alpha (x,y)=\{\begin{array}{ll}1,& \text{if}(x,y)\in E(G),\\ 0,& \text{otherwise.}\end{array}$ At first we prove that the mappings $(S,T)$ are *α*-admissible. Let $x,y\in \overline{B({x}_{0},r)}$ with $\alpha (x,y)\ge 1$, then $(x,y)\in E(G)$. As $(S,T)$ are *ψ*-graphic contractive mappings, we have $(Sx,Ty)\in E(G)$ and $(Tx,Sy)\in E(G)$. That is, $\alpha (Sx,Ty)\ge 1$ and $\alpha (Tx,Sy)\ge 1$. Thus *S*, *T* are *α*-admissible mappings. From (i) there exists ${x}_{0}$ such that $({x}_{0},S{x}_{0})\in E(G)$. That is, $\alpha ({x}_{0},S{x}_{0})\ge 1$.

*S*,

*T*are

*ψ*-graphic contractive mappings, ${d}_{l}(Sx,Ty)\le \psi ({d}_{l}(x,y))$. That is,

Let $\{{x}_{n}\}\subset \overline{B({x}_{0},r)}$ with ${x}_{n}\to x$ as $n\to \mathrm{\infty}$ and $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}$. Then $({x}_{n},{x}_{n+1})\in E(G)$ for all $n\in \mathbb{N}$ and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$. So by (ii) we have $({x}_{n},x)\in E(G)$ for all $n\in \mathbb{N}$. That is, $\alpha ({x}_{n},x)\ge 1$. Hence, all conditions of Corollary 2.3 are satisfied and *S* and *T* have a common fixed point.

Theorem 3.2(2^{
o
}) [24] and Theorem 2.3(2) [12] are extended to *ψ*-graphic contractive pair defined on a dislocated metric space as follows. □

**Theorem 3.4**

*Let*$(X,{d}_{l})$

*be a complete dislocated metric space endowed with a graph*

*G*

*and*$S,T:X\to X$

*be*

*ψ*-

*graphic contractive mappings and*${x}_{0}\in \overline{B({x}_{0},r)}$.

*Suppose that the following assertions hold*:

- (i)
$({x}_{0},S{x}_{0})\in E(G)$

*and*${\sum}_{i=0}^{j}{\psi}^{i}({d}_{l}({x}_{0},S{x}_{0}))\le r$*for all*$j\in \mathbb{N}$;

(iis) $(x,z)\in E(G)$ *and* $(z,y)\in E(G)$ *imply* $(x,y)\in E(G)$ *for all* $x,y,z\in X$, *that is*, $E(G)$ *is a quasi*-*order* [24]*and if* $\{{x}_{n}\}$ *is a sequence in* $\overline{B({x}_{0},r)}$ *such that* $({x}_{n},{x}_{n+1})\in E(G)$ *for all* $n\in \mathbb{N}$ *and* ${x}_{n}\to x$ *as* $n\to +\mathrm{\infty}$, *then there is a subsequence* $\{{x}_{{k}_{n}}\}$ *with* $({x}_{{k}_{n}},x)\in E(G)$ *for all* $n\in \mathbb{N}$.

*Then* *S*, *T* *have a common fixed point*.

*Proof* Condition (iis) implies that of (ii) in Theorem 3.3 (see Remark 3.1 [24]). Now the conclusion follows from Theorem 3.3. □

**Corollary 3.5**

*Let*$(X,{d}_{l})$

*be a complete dislocated metric space endowed with a graph*

*G*

*and*$S,T:X\to X$

*be two mappings and*${x}_{0}\in \overline{B({x}_{0},r)}$.

*Suppose that the following assertions hold*:

- (i)
$(S,T)$

*are**G*-*contractive mappings*; - (ii)
$({x}_{0},S{x}_{0})\in E(G)$

*and*${d}_{l}({x}_{0},S{x}_{0})\le (1-k)r$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in*$\overline{B({x}_{0},r)}$*such that*$({x}_{n},{x}_{n+1})\in E(G)$*for all*$n\in \mathbb{N}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*then*$({x}_{n},x)\in E(G)$*for all*$n\in \mathbb{N}$.

*Then* *S* *and* *T* *have a common fixed point*.

**Corollary 3.6**

*Let*$(X,{d}_{l})$

*be a complete dislocated metric space endowed with a graph*

*G*

*and*$S:X\to X$

*be a mapping and*${x}_{0}\in \overline{B({x}_{0},r)}$.

*Suppose that the following assertions hold*:

- (i)
*S**is Banach**G*-*contraction on*$\overline{B({x}_{0},r)}$; - (ii)
$({x}_{0},S{x}_{0})\in E(G)$

*and*${d}_{l}({x}_{0},S{x}_{0})\le (1-k)r$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in*$\overline{B({x}_{0},r)}$*such that*$({x}_{n},{x}_{n+1})\in E(G)$*for all*$n\in \mathbb{N}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*then*$({x}_{n},x)\in E(G)$*for all*$n\in \mathbb{N}$.

*Then* *S* *has a fixed point*.

**Corollary 3.7**

*Let*$(X,{d}_{l})$

*be a complete dislocated metric space endowed with a graph*

*G*

*and*$S:X\to X$

*be a mapping*.

*Suppose that the following assertions hold*:

- (i)
*S**is Banach**G*-*contraction on**X**and there is*${x}_{0}\in X$*such that*$({x}_{0},S{x}_{0})\in E(G)$; - (ii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$({x}_{n},{x}_{n+1})\in E(G)$*for all*$n\in \mathbb{N}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*then*$({x}_{n},x)\in E(G)$*for all*$n\in \mathbb{N}$.

*Then* *S* *has a fixed point*.

The study of existence of fixed points in partially ordered sets has been initiated by Ran and Reurings [28] with applications to matrix equations. Agarwal, *et al.* [3, 4], Bhaskar and Lakshmikantham [10], Ciric *et al.* [13] and Hussain *et al.* [22, 23] presented some new results for nonlinear contractions in partially ordered metric spaces and noted that their theorems can be used to investigate a large class of problems. Roldán *et al.* [30] and Harandi *et al.* [6] proved some results in preordered metric spaces which is a generalization of partially ordered metric spaces. Here as an application of our results we deduce some new common fixed point results in preordered dislocated metric spaces.

Recall that if $(X,\u2aaf)$ is a preordered set and $T:X\to X$ is such that for $x,y\in X$, with $x\u2aafy$ implies $Tx\u2aafTy$, then the mapping *T* is said to be nondecreasing. If for $x,y\in X$, with $x\u2aafy$ implies $Sx\u2aafTy$ and $Tx\u2aafSy$, then the pair $(S,T)$ is called jointly nondecreasing.

*X*be a non-empty set. Then $(X,{d}_{l},\u2aaf)$ is called a preordered dislocated metric space if ${d}_{l}$ is a dislocated metric on

*X*and ⪯ is a preorder on

*X*. Let $(X,{d}_{l},\u2aaf)$ be a preordered dislocated metric space. Define the graph

*G*by

For this graph, the first condition in Definition 3.2 means *S*, *T* are jointly nondecreasing with respect to this order. From Theorems 3.3-Corollary 3.7 we derive the following important results in preordered dislocated metric spaces.

**Theorem 3.8**

*Let*$(X,{d}_{l},\u2aaf)$

*be a preordered complete dislocated metric space and let the pair*$(S,T)$

*of self*-

*maps of*

*X*

*be jointly nondecreasing and*${x}_{0}\in \overline{B({x}_{0},r)}$.

*Suppose that the following assertions hold*:

- (i)
*for all*$x,y\in \overline{B({x}_{0},r)}$,*with*$x\u2aafy\u27f9{d}_{l}(Sx,Ty)\le \psi ({d}_{l}(x,y))$; - (ii)
${x}_{0}\u2aafS{x}_{0}$

*and*${\sum}_{i=0}^{j}{\psi}^{i}({d}_{l}({x}_{0},S{x}_{0}))\le r$*for all*$j\in \mathbb{N}$; - (iii)
*if*$\{{x}_{n}\}$*is a nondecreasing sequence in*$\overline{B({x}_{0},r)}$*such that*${x}_{n}\to x\in \overline{B({x}_{0},r)}$*as*$n\to +\mathrm{\infty}$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}$.

*Then* *S* *and* *T* *have a common fixed point*.

**Corollary 3.9**

*Let*$(X,{d}_{l},\u2aaf)$

*be a preordered complete dislocated metric space and let the pair*$(S,T)$

*of self*-

*maps of*

*X*

*be jointly nondecreasing and*${x}_{0}\in \overline{B({x}_{0},r)}$.

*Suppose that the following assertions hold*:

- (i)
*there exists*$k\in [0,1)$*such that*${d}_{l}(Sx,Ty)\le k{d}_{l}(x,y)$*for all*$x,y\in \overline{B({x}_{0},r)}$*with*$x\u2aafy$; - (ii)
${x}_{0}\u2aafS{x}_{0}$

*and*${d}_{l}({x}_{0},S{x}_{0})\le (1-k)r$; - (iii)
*if*$\{{x}_{n}\}$*is a nondecreasing sequence in*$\overline{B({x}_{0},r)}$*such that*${x}_{n}\to x\in \overline{B({x}_{0},r)}$*as*$n\to +\mathrm{\infty}$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}$.

*Then* *S* *and* *T* *have a common fixed point*.

**Corollary 3.10**

*Let*$(X,{d}_{l},\u2aaf)$

*be a preordered complete dislocated metric space and let the pair*$(S,T)$

*of self*-

*maps of*

*X*

*be jointly nondecreasing*.

*Suppose that the following assertions hold*:

- (i)
*there exists*$k\in [0,1)$*such that*${d}_{l}(Sx,Ty)\le k{d}_{l}(x,y)$*for all*$x,y\in X$*with*$x\u2aafy$; - (ii)
${x}_{0}\u2aafS{x}_{0}$;

- (iii)
*if*$\{{x}_{n}\}$*is a nondecreasing sequence in**X**such that*${x}_{n}\to x\in X$*as*$n\to +\mathrm{\infty}$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}$.

*Then* *S* *and* *T* *have a common fixed point*.

**Corollary 3.11**

*Let*$(X,{d}_{l},\u2aaf)$

*be a preordered complete dislocated metric space and*$S:X\to X$

*be a nondecreasing map and*${x}_{0}\in \overline{B({x}_{0},r)}$.

*Suppose that the following assertions hold*:

- (i)
*there exists*$k\in [0,1)$*such that*${d}_{l}(Sx,Sy)\le k{d}_{l}(x,y)$*for all*$x,y\in \overline{B({x}_{0},r)}$*with*$x\u2aafy$; - (ii)
${x}_{0}\u2aafS{x}_{0}$

*and*${d}_{l}({x}_{0},S{x}_{0})\le (1-k)r$; - (iii)
*if*$\{{x}_{n}\}$*is a nondecreasing sequence in*$\overline{B({x}_{0},r)}$*such that*${x}_{n}\to x\in \overline{B({x}_{0},r)}$*as*$n\to +\mathrm{\infty}$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}$.

*Then* *S* *has a fixed point*.

**Corollary 3.12**

*Let*$(X,{d}_{l},\u2aaf)$

*be a preordered complete dislocated metric space and*$S:X\to X$

*be a nondecreasing map*.

*Suppose that the following assertions hold*:

- (i)
*there exists*$k\in [0,1)$*such that*${d}_{l}(Sx,Sy)\le k{d}_{l}(x,y)$*for all*$x,y\in X$*with*$x\u2aafy$; - (ii)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafS{x}_{0}$; - (iii)
*if*$\{{x}_{n}\}$*is a nondecreasing sequence in**X**such that*${x}_{n}\to x\in X$*as*$n\to +\mathrm{\infty}$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}$.

*Then* *S* *has a fixed point*.

**Corollary 3.13** [27]

*Let*$(X,d,\u2aaf)$

*be a preordered complete metric space and*$S:X\to X$

*be a nondecreasing mapping such that*

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

*with*$x\u2aafy$

*where*$0\le k<1$.

*Suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafS{x}_{0}$; - (ii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*${x}_{n}\u2aaf{x}_{n+1}$*for all*$n\in \mathbb{N}$*and*${x}_{n}\to x$*as*$n\to +\mathrm{\infty}$,*then*${x}_{n}\u2aafx$*for all*$n\in \mathbb{N}$.

*Then* *S* *has a fixed point*.

**Remark 3.14** We can similarly obtain partial metric and preordered partial metric versions of all results proved here which provide new results in the literature.

## Declarations

### Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.

## Authors’ Affiliations

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