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# Generalized fixed point theorems for multi-valued *α*-*ψ*-contractive mappings

- Nawab Hussain
^{1}Email author, - Jamshaid Ahmad
^{2}and - Akbar Azam
^{2}

**2014**:348

https://doi.org/10.1186/1029-242X-2014-348

© Hussain et al.; licensee Springer. 2014

**Received:**14 May 2014**Accepted:**9 August 2014**Published:**4 September 2014

## Abstract

The aim of this paper is to establish certain new fixed point results for multi-valued as well as single-valued maps satisfying an *α*-*ψ*-contractive conditions in complete metric space. As an application, we derive some new fixed point theorems for *ψ*-graphic contractions defined on a metric space endowed with a graph as well as an ordered metric space. The presented results complement and extend some very recent results proved by Asl *et al.* (Fixed Point Theory Appl. 2012:212, 2012) and Samet *et al.* (Nonlinear Anal. 75:2154-2165, 2012) as well as other theorems given by Hussain *et al.* (Fixed Point Theory Appl. 2013:212, 2013). Some comparative examples are constructed which illustrate the superiority of our results to the existing ones in the literature.

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

## Keywords

- metric space
- fixed point
*α*-*ψ*-contraction

## 1 Introduction

In metric fixed point theory the contractive conditions on underlying functions play an important role for finding solutions of fixed point problems. The Banach contraction principle [1] is a fundamental result in metric fixed point theory. Over the years, it has been generalized in different directions by several mathematicians (see [1–25]). In particular, there has been a number of studies involving altering distance functions which alter the distance between two points in a metric space. In 2012, Samet *et al.* [25] introduced the concepts of *α*-*ψ*-contractive and *α*-admissible mappings and established various fixed point theorems for such mappings in complete metric spaces.

Denote with Ψ the family of 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*.

The following lemma is well known.

**Lemma 1**

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

*then the following hold*:

- (i)
${({\psi}^{n}(t))}_{n\in \mathbb{N}}$

*converges to*0*as*$n\to \mathrm{\infty}$*for all*$t\in (0,+\mathrm{\infty})$; - (ii)
$\psi (t)<t$

*for all*$t>0$; - (iii)
$\psi (t)=0$

*iff*$t=0$.

Samet *et al.* [25] defined the notion of *α*-admissible mappings as follows.

**Definition 2**Let

*T*be a self-mapping on

*X*and $\alpha :X\times X\to [0,+\mathrm{\infty})$ be a function. We say that

*T*is a

*α*-admissible mapping if

**Theorem 3** [25]

*Let*$(X,d)$

*be a complete metric space and*

*T*

*be*

*α*-

*admissible mapping*.

*Assume that*

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

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

*Also*,

*suppose that*

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

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

Afterwards, Asl *et al.* [21] generalized these notions by introducing the concepts of ${\alpha}_{\ast}$-*ψ*-contractive multifunctions, and of ${\alpha}_{\ast}$-admissibility, and they obtained some fixed point results for these multifunctions.

**Definition 4** [21]

*T*is called ${\alpha}_{\ast}$-

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

for all $x,y\in X$, where *H* is the Hausdorff generalized metric, ${\alpha}_{\ast}(A,B)=inf\{\alpha (a,b):a\in A,b\in B\}$ and ${2}^{X}$ denotes the family of all nonempty subsets of *X*.

**Definition 5** [21]

Let $(X,d)$ be a metric space, $T:X\to {2}^{X}$ be a given closed-valued multifunction and $\alpha :X\times X\to [0,+\mathrm{\infty})$. We say that *T* is called ${\alpha}_{\ast}$-admissible whenever $\alpha (x,y)\ge 1$ implies that ${\alpha}_{\ast}(Tx,Ty)\ge 1$.

Very recently Hussain *et al.* [12] modified the notions of ${\alpha}_{\ast}$-admissible and ${\alpha}_{\ast}$-*ψ*-contractive mappings as follows:

**Definition 6**Let $T:X\to {2}^{X}$ be a multifunction, $\alpha ,\eta :X\times X\to {\mathbb{R}}_{+}$ be two functions where

*η*is bounded. We say that

*T*is ${\alpha}_{\ast}$-admissible mapping with respect to

*η*if

If $\eta (x,y)=1$ for all $x,y\in X$, then this definition reduces to Definition 5. In the case $\alpha (x,y)=1$ for all $x,y\in X$, *T* is called ${\eta}_{\ast}$-subadmissible mapping.

Hussain *et al.* [12] proved following generalization of the above mentioned results of [21].

**Theorem 7**

*Let*$(X,d)$

*be a complete metric space and*$T:X\to {2}^{X}$

*be a*${\alpha}_{\ast}$-

*admissible with respect to*

*η*

*and the closed*-

*valued multifunction on*

*X*.

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

*Also suppose that the following assertions hold*:

- (i)
*there exist*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$*such that*$\alpha ({x}_{0},{x}_{1})\ge \eta ({x}_{0},{x}_{1})$; - (ii)
*for a sequence*$\{{x}_{n}\}\subset X$*converging to*$x\in X$*and*$\alpha ({x}_{n},{x}_{n+1})\ge \eta ({x}_{n},{x}_{n+1})$*for all*$n\in \mathbb{N}$,*we have*$\alpha ({x}_{n},x)\ge \eta ({x}_{n},x)$*for all*$n\in \mathbb{N}$.

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

For more details on *α*-*ψ*-contractions and fixed point theory, we refer the reader to [3, 6, 10, 13, 14, 22, 23, 26–29].

The aim of this paper is to unify the concepts of *α*-*ψ*-contractive type mappings and establish some new fixed point theorems in complete metric spaces for such mappings.

Let $(X,d)$ be a complete metric space, ${x}_{0}\in X$ and $r>0$. We denote by $B({x}_{0},r)=\{x\in X:d({x}_{0},x)<r\}$ the open ball with center ${x}_{0}$ and radius *r* and by $\overline{B({x}_{0},r)}=\{x\in X:d({x}_{0},x)\le r\}$ the closed ball with center ${x}_{0}$ and radius *r*.

The following lemmas of Nadler will be needed in the sequel.

**Lemma 8** [19]

*Let* *A* *and* *B* *be nonempty*, *closed and bounded subsets of a metric space* $(X,d)$ *and* $0<h\in \mathbb{R}$. *Then*, *for every* $b\in B$, *there exists* $a\in A$ *such that* $d(a,b)\le H(A,B)+h$.

**Lemma 9** [4]

*Let* $(X,d)$ *be a metric space and* *B* *be nonempty*, *closed subsets of* *X* *and* $q>1$. *Then*, *for each* $x\in X$ *with* $d(x,B)>0$ *and* $q>1$, *there exists* $b\in B$ *such that* $d(x,b)<qd(x,B)$.

## 2 Main result

The following result, regarding the existence of the fixed point of the mapping satisfying an *α*-*ψ*-contractive condition on the closed ball, is very useful in the sense that it requires the contractiveness of the mapping only on the closed ball instead of the whole space.

**Theorem 10**

*Let*$(X,d)$

*be a complete metric space and*$T:X\to {2}^{X}$

*be an*${\alpha}_{\ast}$-

*admissible and closed*-

*valued multifunction on*

*X*.

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

*for all*$x,y\in \overline{B({x}_{0},r)}$

*and for*${x}_{0}\in X$,

*there exists*${x}_{1}\in T{x}_{0}$

*such that*

*for all*$n\in \mathbb{N}$

*and*$r>0$.

*Also suppose that the following assertions hold*:

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

*for*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$; - (ii)
*for a sequence*$\{{x}_{n}\}$*in*$\overline{B({x}_{0},r)}$*converging to*$x\in \overline{B({x}_{0},r)}$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}$.

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

*Proof*Since $\alpha ({x}_{0},{x}_{1})\ge 1$ and

*T*is ${\alpha}_{\ast}$-admissible, so ${\alpha}_{\ast}(T{x}_{0},T{x}_{1})\ge 1$. From (2.2), we get

*n*and

*T*is ${\alpha}_{\ast}$-admissible with respect to

*η*, so ${\alpha}_{\ast}(T{x}_{n},T{x}^{\ast})\ge 1$ for all

*n*. Then

Taking the limit as $n\to \mathrm{\infty}$ in (2.5), we get $d({x}^{\ast},T{x}^{\ast})=0$. Thus ${x}^{\ast}\in T{x}^{\ast}$. □

**Example 11**Let $X=[0,\mathrm{\infty})$ and $d(x,y)=|x-y|$. Define the multi-valued mapping $T:X\to {2}^{X}$ by

*T*is an

*α*-

*ψ*-contractive mapping with $\psi (t)=\frac{t}{2}$. Now

Put ${x}_{0}=\frac{1}{2}$ and ${x}_{1}=\frac{1}{4}$. Then $\alpha ({x}_{0},{x}_{1})\ge 1$. Then *T* has a fixed point 0.

Similarly we can deduce the following corollaries.

**Corollary 12**

*Let*$(X,d)$

*be a complete metric space and*$T:X\to {2}^{X}$

*be an*${\alpha}_{\ast}$-

*admissible and closed*-

*valued multifunction on*

*X*.

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

*we have*

*for all*$x,y\in \overline{B({x}_{0},r)}$

*and for*${x}_{0}\in X$,

*there exists*${x}_{1}\in T{x}_{0}$

*such that*

*for all*$n\in \mathbb{N}$

*and*$r>0$.

*Also suppose that the following assertions hold*:

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

*for*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$; - (ii)
*for a sequence*$\{{x}_{n}\}$*in*$\overline{B({x}_{0},r)}$*converging to*$x\in \overline{B({x}_{0},r)}$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}$.

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

**Corollary 13**

*Let*$(X,d)$

*be a complete metric space and*$T:X\to {2}^{X}$

*be an*${\alpha}_{\ast}$-

*admissible and closed*-

*valued multifunction on*

*X*.

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

*we have*

*for all*$x,y\in \overline{B({x}_{0},r)}$

*and*$l>0$

*and for*${x}_{0}\in X$,

*there exists*${x}_{1}\in T{x}_{0}$

*such that*

*for all*$n\in \mathbb{N}$

*and*$r>0$.

*Also suppose that the following assertions hold*:

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

*for*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$; - (ii)
*for a sequence*$\{{x}_{n}\}$*in*$\overline{B({x}_{0},r)}$*converging to*$x\in \overline{B({x}_{0},r)}$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}$.

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

**Theorem 14**

*Let*$(X,d)$

*be a complete metric space and*$T:X\to {2}^{X}$

*be an*${\alpha}_{\ast}$-

*admissible and closed*-

*valued multifunction on*

*X*.

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

*we have*

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

*Also suppose that the following assertions hold*:

- (i)
*there exist*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$*with*$\alpha ({x}_{0},{x}_{1})\ge 1$; - (ii)
*for a sequence*$\{{x}_{n}\}$*in**X**converging to*$x\in X$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}$.

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

*Proof*Since $\alpha ({x}_{0},{x}_{1})\ge 1$ and

*T*is ${\alpha}_{\ast}$-admissible, so ${\alpha}_{\ast}(T{x}_{0},T{x}_{1})\ge 1$. If ${x}_{0}={x}_{1}$, then we have nothing to prove. Let ${x}_{0}\ne {x}_{1}$. If ${x}_{1}\in T{x}_{1}$, then ${x}_{1}$ is a fixed point of

*T*. Assume that ${x}_{1}\notin T{x}_{1}$, then from (2.7), we get

*T*is ${\alpha}_{\ast}$-admissible, so ${\alpha}_{\ast}(T{x}_{1},T{x}_{2})\ge 1$. If ${x}_{2}\in T{x}_{2}$, then ${x}_{2}$ is fixed point of

*T*. Assume that ${x}_{2}\notin T{x}_{2}$. Then from (2.7), we get

*T*is ${\alpha}_{\ast}$-admissible, so ${\alpha}_{\ast}(T{x}_{2},T{x}_{3})\ge 1$. If ${x}_{3}\in T{x}_{3}$, then ${x}_{3}$ is fixed point of

*T*. Assume that ${x}_{3}\notin T{x}_{3}$. From (2.7), we have

*X*such that ${x}_{n}\in T{x}_{n-1}$ and ${x}_{n}\ne {x}_{n-1}$, and

*n*. Now, for each $m>n$, we have

*X*. Since

*X*is complete, there exists ${x}^{\ast}\in X$ such that ${x}_{n}\u27f6{x}^{\ast}$ as $n\u27f6\mathrm{\infty}$. We now show that ${x}^{\ast}\in T{x}^{\ast}$. Since $\alpha ({x}_{n},{x}^{\ast})\ge 1$ for all

*n*and

*T*is ${\alpha}_{\ast}$-admissible, so ${\alpha}_{\ast}(T{x}_{n},T{x}^{\ast})\ge 1$ for all

*n*. Then

and taking the limit as $n\to \mathrm{\infty}$, we get $d({x}^{\ast},T{x}^{\ast})=0$. Thus ${x}^{\ast}\in T{x}^{\ast}$. □

**Example 15**Let $X=[0,1]$ and $d(x,y)=|x-y|$. Define $T:X\to {2}^{X}$ by $Tx=[0,\frac{x}{10}]$ for all $x\in X$ and

*T*is ${\alpha}^{\ast}$-admissible. Now, for

*x*,

*y*and $x\le y$, it is easy to check that

where $\psi (t)=\frac{t}{5}$, for all $t\ge 0$. Put ${x}_{0}=1$ and ${x}_{1}=\frac{1}{2}$. Then $\alpha ({x}_{0},{x}_{1})=2>1$. Then *T* has fixed point 0.

**Corollary 16**

*Let*$(X,d)$

*be a complete metric space and*$T:X\to {2}^{X}$

*be an*${\alpha}_{\ast}$-

*admissible and closed*-

*valued multifunction on*

*X*.

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

*we have*

*where*

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

*Also suppose that the following assertions hold*:

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

*for*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$; - (ii)
*for a sequence*$\{{x}_{n}\}$*in**X**converging to*$x\in X$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}$.

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

**Corollary 17**

*Let*$(X,d)$

*be a complete metric space and*$T:X\to {2}^{X}$

*be an*${\alpha}_{\ast}$-

*admissible and closed*-

*valued multifunction on*

*X*.

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

*we have*

*where*

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

*and*$l>0$.

*Also suppose that the following assertions hold*:

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

*for*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$; - (ii)
*for a sequence*$\{{x}_{n}\}$*in**X**converging to*$x\in X$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}$.

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

If *T* is single-valued in Theorem 14, we obtain the following fixed point results.

**Theorem 18**

*Let*$(X,d)$

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

*be an*

*α*-

*admissible mapping*.

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

*we have*

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

*Also suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*with*$\alpha ({x}_{0},T{x}_{0})\ge 1$; - (ii)
*for a sequence*$\{{x}_{n}\}$*in**X**converging to*$x\in X$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}$.

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

**Corollary 19**

*Let*$(X,d)$

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

*be an*

*α*-

*admissible mapping*.

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

*we have*

*where*

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

*Also suppose that the following assertions hold*:

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

*for some*${x}_{0}\in X$; - (ii)
*for a sequence*$\{{x}_{n}\}$*in**X**converging to*$x\in X$*and*$\alpha ({x}_{n},{x}_{n+1})\ge 1$*for all*$n\in \mathbb{N}$,*we have*$\alpha ({x}_{n},x)\ge 1$*for all*$n\in \mathbb{N}$.

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

Now, we give the following result about a fixed point of self-maps on complete metric spaces.

**Theorem 20**

*Let*$(X,d)$

*be a complete metric space*, $\alpha :X\times X\to [0,+\mathrm{\infty})$

*be a mapping*, $\psi \in \mathrm{\Psi}$

*and*

*T*

*be a self*-

*mapping on*

*X*

*such that*

*for all* $x,y\in X$. *Suppose that T is* *α*-*admissible and there exist* ${x}_{0}\in X$ *and* ${x}_{1}\in T{x}_{0}$ *with* $\alpha ({x}_{0},T{x}_{0})\ge 1$. *If* *T* *is continuous*. *Then* *T* *has a unique fixed point*.

*Proof*Take ${x}_{0}\in X$ such that $\alpha ({x}_{0},T{x}_{0})\ge 1$, and define the sequence $\{{x}_{n}\}$ in

*X*by ${x}_{n+1}=T{x}_{n}$ for all $n\ge 0$. If ${x}_{n}={x}_{n+1}$ for some

*n*, then ${x}^{\ast}={x}_{n}$ is a fixed point of

*T*. Assume that ${x}_{n}\ne {x}_{n+1}$ for all

*n*. Since

*T*is

*α*-admissible, so it is easy to check that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all natural numbers

*n*. Thus for each natural number

*n*, we have

*ψ*is nondecreasing, so we have

*n*. It is easy to check that $\{{x}_{n}\}$ is a Cauchy sequence. Since

*X*is complete, so there exists ${x}^{\ast}\in X$ such that ${x}_{n}\to {x}^{\ast}$. Further the continuity of

*T*implies that

*T*in

*X*. Now, if there exists another point $u\ne {x}^{\ast}$ in

*X*such that $Tu=u$, then

a contradiction. Hence ${x}^{\ast}$ is a unique fixed point of *T* in *X*. □

**Example 21**Let $X=[0,\mathrm{\infty})$ and $d(x,y)=|x-y|$. Define $T:X\to X$ by $Tx=x+1$ whenever $x,y\in [0,1]$, $Tx=\frac{4}{3}$ whenever $x,y\in (1,2)$ and $Tx={x}^{2}+3x+2$ whenever $x\in [2,\mathrm{\infty})$. Also define the mappings $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ by $\psi (t)=\frac{t}{3}$ and

for all $x,y\in X$ and $\frac{4}{3}$ is unique fixed point of the mapping *T*.

## 3 Fixed point results for graphic contractions

Consistent with Jachymski [15], let $(X,d)$ be a metric space and Δ denote 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 [15]) 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 *N* ($N\in \mathbb{N}$) is a sequence ${\{{x}_{i}\}}_{i=0}^{N}$ of $N+1$ vertices such that ${x}_{0}=x$, ${x}_{N}=y$ and $({x}_{n-1},{x}_{n})\in E(G)$ for $i=1,\dots ,N$. 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 [7, 9, 13, 15]).

**Definition 22** [15]

*G*-contraction or simply

*G*-contraction if

*T*preserves edges of

*G*,

*i.e.*,

*T*decreases weights of edges of

*G*in the following way:

**Definition 23** [15]

*G*-continuous, if given $x\in X$ and the sequence $\{{x}_{n}\}$

**Theorem 24**

*Let*$(X,d)$

*be a complete metric space endowed with a graph*

*G*

*and*

*T*

*be a self*-

*mapping on*

*X*.

*Suppose the following assertions hold*:

- (i)
$\mathrm{\forall}x,y\in X$, $(x,y)\in E(G)\Rightarrow (T(x),T(y))\in E(G)$;

- (ii)
*there exists*${x}_{0}\in X$*such that*$({x}_{0},T{x}_{0})\in E(G)$; - (iii)
*there exists*$\psi \in \mathrm{\Psi}$*such that*$d(Tx,Ty)\le \psi (R(x,y))$

*for all*$(x,y)\in E(G)$

*where*

- (iv)
*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* *T* *has a fixed point*.

*Proof* Define, $\alpha :{X}^{2}\to [0,+\mathrm{\infty})$ by $\alpha (x,y)=\{\begin{array}{ll}1,& \text{if}(x,y)\in E(G),\\ 0,& \text{otherwise}.\end{array}$ First we prove that *T* is an *α*-admissible mapping. Let, $\alpha (x,y)\ge 1$, then $(x,y)\in E(G)$. From (i), we have $(Tx,Ty)\in E(G)$. That is, $\alpha (Tx,Ty)\ge 1$. Thus *T* is an *α*-admissible mapping. From (ii) there exists ${x}_{0}\in X$ such that $({x}_{0},T{x}_{0})\in E(G)$. That is, $\alpha ({x}_{0},T{x}_{0})\ge 1$. If $(x,y)\in E(G)$, then $(Tx,Ty)\in E(G)$ and hence $\alpha (Tx,Ty)=1$. Thus, from (iii) we have $\alpha (Tx,Ty)d(Tx,Ty)=d(Tx,Ty)\le \psi (M(x,y))$. Condition (iv) implies condition (ii) of Theorem 18. Hence, all conditions of Theorem 18 are satisfied and *T* has a fixed point. □

**Corollary 25**

*Let*$(X,d)$

*be a complete metric space endowed with a graph*

*G*

*and*

*T*

*be a self*-

*mapping on*

*X*.

*Suppose the following assertions hold*:

- (i)
*T**is a Banach**G*-*contraction*; - (ii)
*there exists*${x}_{0}\in X$*such that*$({x}_{0},T{x}_{0})\in E(G)$; - (iii)
*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* *T* *has a fixed point*.

As an application of Theorem 20, we obtain;

**Theorem 26**

*Let*$(X,d)$

*be a complete metric space endowed with a graph*

*G*

*and*

*T*

*be a self*-

*mapping on*

*X*.

*Suppose the following assertions hold*:

- (i)
$\mathrm{\forall}x,y\in X$, $(x,y)\in E(G)\Rightarrow (T(x),T(y))\in E(G)$;

- (ii)
*there exists*${x}_{0}\in X$*such that*$({x}_{0},T{x}_{0})\in E(G)$; - (iii)
*there exists*$\psi \in \mathrm{\Psi}$*such that*$d(Tx,Ty)\le \{\begin{array}{ll}\psi (max\{\frac{d(x,Tx)d(y,Ty)}{d(x,y)},d(x,y)\})& \mathit{\text{for all}}(x,y)\in E(G)\mathit{\text{with}}x\ne y,\\ 0& \mathit{\text{for}}x=y;\end{array}$ - (iv)
*T**is**G*-*continuous*.

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

*G*by

For this graph, condition (i) in Theorem 24 means *T* is nondecreasing with respect to this order [8]. From Theorems 24-26 we derive the following important results in partially ordered metric spaces.

**Theorem 27**

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

*be a complete partially ordered metric space and*

*T*

*be a self*-

*mapping on*

*X*.

*Suppose the following assertions hold*:

- (i)
*T**is nondecreasing map*; - (ii)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (iii)
*there exists*$\psi \in \mathrm{\Psi}$*such that*$d(Tx,Ty)\le \psi (R(x,y))$

*for all*$x\u2aafy$

*where*

- (iv)
*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* *T* *has a fixed point*.

**Corollary 28** [20]

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

*be a complete partially ordered metric space and*$T:X\to X$

*be nondecreasing mapping such that*

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

*with*$x\u2aafy$

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

*Suppose that the following assertions hold*:

- (i)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (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* *T* *has a fixed point*.

**Theorem 29**

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

*be a complete partially ordered metric space and*

*T*

*be a self*-

*mapping on*

*X*.

*Suppose the following assertions hold*:

- (i)
*T**is nondecreasing map*; - (ii)
*there exists*${x}_{0}\in X$*such that*${x}_{0}\u2aafT{x}_{0}$; - (iii)
*there exists*$\psi \in \mathrm{\Psi}$*such that*$d(Tx,Ty)\le \{\begin{array}{ll}\psi (max\{\frac{d(x,Tx)d(y,Ty)}{d(x,y)},d(x,y)\})& \mathit{\text{for all}}x\u2aafy\mathit{\text{with}}x\ne y,\\ 0& \mathit{\text{for}}x=y;\end{array}$ - (iv)
*T**is continuous*.

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

## 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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