# The existence of fixed point theorems for partial *q*-set-valued quasi-contractions in *b*-metric spaces and related results

- Poom Kumam
^{1}and - Wutiphol Sintunavarat
^{2}Email author

**2014**:226

https://doi.org/10.1186/1687-1812-2014-226

© Kumam and Sintunavarat; licensee Springer. 2014

**Received: **2 May 2014

**Accepted: **6 October 2014

**Published: **1 November 2014

## Abstract

In this paper, we present a new type of set-valued mappings called partial *q*-set-valued quasi-contraction mappings and give results as regards fixed points for such mappings in *b*-metric spaces. By providing some examples, we show that our results are real generalizations of the main results of Aydi *et al.* (Fixed Point Theory Appl. 2012:88, 2012) and many results in the literature. We also consider fixed point results for single-valued mapping, fixed point results for set-valued mapping in *b*-metric space endowed with an arbitrary binary relation, and fixed point results in a *b*-metric space endowed with a graph. By using our result, we establish the existence of solution for the following an integral equations: $x(c)=\varphi (c)+{\int}_{a}^{b}K(c,r,x(r))\phantom{\rule{0.2em}{0ex}}dr$, where $b>a\ge 0$, $x\in C[a,b]$ (the set of continuous real functions defined on $[a,b]\subseteq \mathbb{R}$), $\varphi :[a,b]\to \mathbb{R}$, and $K:[a,b]\times [a,b]\times \mathbb{R}\to \mathbb{R}$ are given mappings.

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

## Keywords

*α*-admissible mappingsbinary relationsfixed points

*b*-metric spaces

*q*-set-valued

*α*-quasi-contraction mappings

## 1 Introduction

The Banach contraction principle is a very popular tool of mathematics in solving many problems in several branches of mathematics since it can be observed easily and comfortably. In 1993, Czerwik [1] introduced the concept of *b*-metric spaces and also presented the fixed point theorem for contraction mappings in *b*-metric spaces, that is, we have a generalization of the Banach contraction principle in metric spaces. Afterward, many mathematicians studied fixed point theorems for single-valued and set-valued mappings in *b*-metric spaces (see [2–7] and references therein).

In 2012, Aydi *et al.* [8] extended the concept of *q*-set-valued quasi-contraction mappings in metric spaces due to Amini-Harandi [9] to *b*-metric spaces. They also established the fixed point results for *q*-set-valued quasi-contraction mappings in *b*-metric spaces. Recently, Sintunavarat *et al.* [10] introduced some set-valued mappings called *q*-set-valued *α*-quasi-contraction mappings and obtained fixed point results for such mappings in *b*-metric spaces which are generalization of the results of Aydi *et al.* [8], Amini-Harandi [9] and many works in the literature.

*q*-set-valued quasi-contraction mappings which is the wider class of many classes in this field. As regards this class, we study and obtain fixed point results in

*b*-metric spaces. These results extend, unify and generalize several well-known comparable results in the existing literature. As an application of our results, we prove the fixed point theorems for a single-valued mapping and give an example to show the generality of our result. We also study the fixed point results in a

*b*-metric space endowed with an arbitrary binary relation and endowed with a graph. As applications, we apply our result to the proof of the existence of a solution for the following an integral equation:

where $b>a\ge 0$, $x\in C[a,b]$ (the set of continuous real functions defined on $[a,b]\subseteq \mathbb{R}$), $\varphi :[a,b]\to \mathbb{R}$, and $K:[a,b]\times [a,b]\times \mathbb{R}\to \mathbb{R}$ are given mappings.

## 2 Preliminaries

In this section, we give some notations and basic knowledge in nonlinear analysis and *b*-metric spaces. Throughout this paper, ℝ, ${\mathbb{R}}_{+}$, and ℕ denote the set of real numbers, the set of nonnegative real numbers, and the set of positive integers, respectively.

**Definition 2.1** ([1])

Let *X* be a nonempty set and $s\ge 1$ be a given real number. A functional $d:X\times X\to {\mathbb{R}}_{+}$ is called a *b*-metric if, for all $x,y,z\in X$, the following conditions are satisfied:

(B_{1}) $d(x,y)=0$ if and only if $x=y$;

(B_{2}) $d(x,y)=d(y,x)$;

(B_{3}) $d(x,z)\le s[d(x,y)+d(y,z)]$.

A pair $(X,d)$ is called a *b*-metric space with coefficient *s*.

**Remark 2.2** The result is obtained that any metric space is a *b*-metric space with $s=1$. Thus the class of *b*-metric spaces is larger than the class of metric spaces.

Some examples of *b*-metric spaces are given by Berinde [11], Czerwik [6], Heinonen [12]. Some well-known examples of a *b*-metric which show that the *b*-metric space is a real generalization of metric space are the following.

**Example 2.3**The set of real numbers together with the functional $d:\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}_{+}$,

*b*-metric space with coefficient $s=2$. However, we find that

*d*is not a metric on

*X*since the ordinary triangle inequality is not satisfied. Indeed,

**Example 2.4**Let $(X,d)$ be a metric space and a functional $\rho :\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}_{+}$ defined by $\rho (x,y)={(d(x,y))}^{p}$, where $p>1$ is a fixed real number. We show that

*ρ*is a

*b*-metric with $s={2}^{p-1}$. It is easy to see that conditions (B

_{1}) and (B

_{2}) are satisfied. If $1<p<\mathrm{\infty}$, then the convexity of the function $f(x)={x}^{p}$ ($x>0$) implies the following inequality:

Consequently, condition (B_{3}) is also satisfied and thus *ρ* is a *b*-metric on *X*.

**Example 2.5**The set ${l}_{p}(\mathbb{R})$ with $0<p<1$, where

for each $x=\{{x}_{n}\},y=\{{y}_{n}\}\in {l}_{p}(\mathbb{R})$, is a *b*-metric space with coefficient $s={2}^{\frac{1}{p}}>1$. We see that the above result also holds for the general case ${l}_{p}(X)$ with $0<p<1$, where *X* is a Banach space.

**Example 2.6**Let

*p*be a given real number in the interval $(0,1)$. The space ${L}_{p}[0,1]$ of all real functions $x(t)$, $t\in [0,1]$ such that ${\int}_{0}^{1}{|x(t)|}^{p}\phantom{\rule{0.2em}{0ex}}dt<1$, together with the functional $d:{L}_{p}[0,1]\times {L}_{p}[0,1]\to {\mathbb{R}}_{+}$,

is a *b*-metric space with constant $s={2}^{\frac{1}{p}}$.

**Example 2.7**Let $X=\{0,1,2\}$ and a functional $d:X\times X\to {\mathbb{R}}_{+}$ be defined by

*m*is given real number such that $m\ge 2$. It easy to see that

for all $x,y,z\in X$. Therefore, $(X,d)$ is a *b*-metric space with coefficient $s=m/2$. We find that the ordinary triangle inequality does not hold if $m>2$ and then $(X,d)$ is not a metric space.

Next, we give the concepts of convergence, compactness, closedness, and completeness in a *b*-metric space.

**Definition 2.8** ([4])

*b*-metric space. The sequence $\{{x}_{n}\}$ in

*X*is called:

- (1)
*convergent*if and only if there exists $x\in X$ such that $d({x}_{n},x)\to 0$ as $n\to \mathrm{\infty}$. In this case, we write ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$. - (2)
*Cauchy*if and only if $d({x}_{n},{x}_{m})\to 0$ as $m,n\to \mathrm{\infty}$.

**Remark 2.9**In a

*b*-metric space $(X,d)$ the following assertions hold:

- (1)
a convergent sequence has a unique limit;

- (2)
each convergent sequence is Cauchy;

- (3)
in general a functional

*b*-metric $d:X\times X\to {\mathbb{R}}_{+}$ for coefficient $s>1$ is not jointly continuous in all its variables.

The following example is an example of a *b*-metric which is not continuous.

**Example 2.10** (see [13])

_{1}) and (B

_{2}) are satisfied. Also, for each $x,y,z\in X$, we have

Therefore, $(X,d)$ is a *b*-metric space on *X* with coefficient $s=3$.

*d*is not continuous. Let ${x}_{n}=2n$ for each $n\in \mathbb{N}$. It is easy to see that

that is, ${x}_{n}\to \mathrm{\infty}$, but $d({x}_{n},1)=2\nrightarrow d(\mathrm{\infty},1)$ as $n\to \mathrm{\infty}$. Therefore, *d* is not continuous.

**Definition 2.11** The *b*-metric space $(X,d)$ is *complete* if every Cauchy sequence in *X* converges.

**Definition 2.12** ([4])

*Y*be a nonempty subset of a

*b*-metric space

*X*. The closure $\overline{Y}$ of

*Y*is the set of limits of all convergent sequences of points in

*Y*,

*i.e.*,

**Definition 2.13** ([4])

*b*-metric space. A subset $Y\subseteq X$ is called:

- (1)
*closed*if and only if for each sequence $\{{x}_{n}\}$ in*Y*which converges to an element*x*, we have $x\in Y$ (*i.e.*$Y=\overline{Y}$); - (2)
*compact*if and only if for every sequence of element in*Y*there exists a subsequence that converges to an element in*Y*; - (3)
*bounded*if and only if $\delta (Y):=sup\{d(a,b)\mid a,b\in Y\}<\mathrm{\infty}$.

*b*-metric space $(X,d)$:

Next, we give the concept of generalized functionals on a *b*-metric space $(X,d)$.

**Definition 2.14**Let $(X,d)$ be a

*b*-metric space.

- (1)The functional $D:\mathcal{P}(X)\times \mathcal{P}(X)\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ is said to be
*a gap functional*if and only if it is defined by$D(A,B)=\{\begin{array}{ll}inf\{d(a,b)\mid a\in A,b\in B\},& A\ne \mathrm{\varnothing}\ne B,\\ 0,& A=\mathrm{\varnothing}=B,\\ +\mathrm{\infty},& \text{otherwise}.\end{array}$In particular, if ${x}_{0}\in X$ then $d({x}_{0},B):=D(\{{x}_{0}\},B)$.

- (2)The functional $\rho :\mathcal{P}(X)\times \mathcal{P}(X)\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ is said to be an
*excess generalized functional*if and only if it is defined by$\rho (A,B)=\{\begin{array}{ll}sup\{d(a,B)\mid a\in A\},& A\ne \mathrm{\varnothing}\ne B,\\ 0,& A=\mathrm{\varnothing},\\ +\mathrm{\infty},& \text{otherwise}.\end{array}$ - (3)The functional $H:\mathcal{P}(X)\times \mathcal{P}(X)\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ is said to be a
*Pompeiu-Hausdorff generalized functional*if and only if it is defined by$H(A,B)=\{\begin{array}{ll}max\{\rho (A,B),\rho (B,A)\},& A\ne \mathrm{\varnothing}\ne B,\\ 0,& A=\mathrm{\varnothing},\\ +\mathrm{\infty},& \text{otherwise}.\end{array}$

**Remark 2.15**For

*b*-metric space $(X,d)$, the following assertions hold:

- (1)
$({P}_{cp}(X),H)$ is a complete

*b*-metric space provided $(X,d)$ is a complete*b*-metric space; - (2)for each $A,B\in P(X)$ and $x\in A$, we have$d(x,B)\le \rho (A,B)\le H(A,B);$
- (3)for $x\in X$ and $B\in P(X)$, we get$d(x,B)\le d(x,b),$

for all $b\in B$.

The following lemmas are useful for the proofs in the main result.

**Lemma 2.16** ([6])

*Let*$(X,d)$

*be a*

*b*-

*metric space*.

*Then*

*for all*$x\in X$

*and*$A,B\in P(X)$.

*In particular*,

*we have*

*for all* $x,y\in X$ *and* $A\in P(X)$.

**Lemma 2.17** ([6])

*Let* $(X,d)$ *be a* *b*-*metric space and* $A,B\in {P}_{b,cl}(X)$. *Then for each* $\u03f5>0$ *and*, *for all* $b\in B$, *there exists* $a\in A$ *such that* $d(a,b)\le H(A,B)+\u03f5$.

**Lemma 2.18** ([6])

*Let*$(X,d)$

*be a*

*b*-

*metric space*.

*For*$A\in {P}_{b,cl}(X)$

*and*$x\in X$,

*we have*

**Lemma 2.19** ([14])

*Let*$(X,d)$

*be a*

*b*-

*metric space with coefficient*$s\ge 1$

*and*$\{{x}_{n}\}$

*be a sequence in*

*X*

*such that*

*for all* $n\in \mathbb{N}$, *where* $0\le \gamma <1$. *Then* $\{{x}_{n}\}$ *is a Cauchy sequence in* *X* *provided that* $s\gamma <1$.

In 2012, Samet *et al.* [15] introduced the concepts of *α*-admissible mapping as follows.

**Definition 2.20** ([15])

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

*t*is

*α*-admissible if

They proved the fixed point results for single-valued mapping as regards this concept and also showed that these results can be utilized to derive fixed point theorems in partially ordered spaces. As an application, they obtain the existence of solutions for ordinary differential equations.

Afterward, Asl *et al.* [16] and Mohammadi *et al.* [17] introduced the concept of ${\alpha}_{\ast}$-admissibility and *α*-admissibility for set-valued mappings as follows.

*X*be a nonempty set, $T:X\to {2}^{X}$, where ${2}^{X}$ is a collection of nonempty subsets of

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

- (1)
*T*is ${\alpha}_{\ast}$-admissible if$\text{for}x,y\in X\text{for which}\alpha (x,y)\ge 1\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}{\alpha}_{\ast}(Tx,Ty)\ge 1,$where ${\alpha}_{\ast}(Tx,Ty):=inf\{\alpha (a,b)\mid a\in Tx,b\in Ty\}$.

- (2)
*T*is*α*-admissible if for each $x\in X$ and $y\in Tx$ with $\alpha (x,y)\ge 1$, we have $\alpha (y,z)\ge 1$, for all $z\in Ty$.

**Remark 2.22** If *T* is ${\alpha}_{\ast}$-admissible, then *T* is also *α*-admissible mapping.

In recent investigations, the fixed point results for single-valued and set-valued mappings via the concepts of being *α*-admissible and ${\alpha}_{\ast}$-admissible occupies a prominent place in many aspects (see [18–25] and references therein).

## 3 Fixed point theorems for partial *q*-set-valued quasi-contraction mappings

In this section, we introduce the partial *q*-set-valued quasi-contraction mapping and obtain the theorem of the existence of a fixed point for such a mapping in *b*-metric spaces.

*X*and the given mapping $\alpha :X\times X\to [0,\mathrm{\infty})$, we use the following notation:

**Definition 3.1**Let $(X,d)$ be a

*b*-metric space and $\alpha :X\times X\to [0,\mathrm{\infty})$ be a given mapping. The set-valued mapping $T:X\to {P}_{b,cl}(X)$ is said to be a

*partial*

*q-set-valued quasi-contraction*if, for all $(x,y)\in X\times X$,

where $0\le q<1$.

Next, we give the main result in this paper.

**Theorem 3.2**

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*$s\ge 1$, $\alpha :X\times X\to [0,\mathrm{\infty})$

*be a given mapping and*$T:X\to {P}_{b,cl}(X)$

*be a partial*

*q*-

*set*-

*valued quasi*-

*contraction*.

*Suppose that the following conditions hold*:

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

*If* $q<\frac{1}{{s}^{2}+s}$, *then* *T* *has a fixed point in* *X*, *that is*, *there exists* $u\in X$ *such that* $u\in Tu$.

*Proof*For $x,y\in X$, we obtain

*T*. Therefore, we suppose that

for all $x,y\in X$.

It follows from $q<\frac{1}{{s}^{2}+s}$ that $\epsilon >0$ and $0<\beta <\frac{1}{{s}^{2}+s}$.

*T*is

*α*-admissible, ${x}_{0}\in X$, and ${x}_{1}\in T{x}_{0}$ such that $\alpha ({x}_{0},{x}_{1})\ge 1$, we get $\alpha ({x}_{1},{x}_{2})\ge 1$ and so $({x}_{1},{x}_{2})\in {\bigwedge}_{\alpha}$. Using Lemma 2.17, there exists ${x}_{3}\in T{x}_{2}$ such that

*T*is a partial

*q*-set-valued quasi-contraction and $({x}_{1},{x}_{2})\in {\bigwedge}_{\alpha}$, we obtain

*X*such that, for each $n\in \mathbb{N}$, we have

where $\gamma :=\frac{\beta s}{1-\beta s}$.

*X*. By the completeness of

*X*, there exists $u\in X$ such that

It follows from $q<\frac{1}{{s}^{2}+s}$ that $q{s}^{2}<1$. From (3.10), we get $d(u,Tu)=0$. Using Lemma 2.18, we have $u\in Tu$, that is, *u* is a fixed point of *T*. This completes the proof. □

**Theorem 3.3**

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*$s\ge 1$, $\alpha :X\times X\to [0,\mathrm{\infty})$

*be a given mapping and*$T:X\to {P}_{b,cl}(X)$

*be a partial*

*q*-

*set*-

*valued quasi*-

*contraction*.

*Suppose that the following conditions hold*:

- (i)
*T**is*${\alpha}_{\ast}$-*admissible*; - (ii)
*there exist*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$*such that*$({x}_{0},{x}_{1})\in {\bigwedge}_{\alpha}$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$({x}_{n},{x}_{n+1})\in {\bigwedge}_{\alpha}$,*for all*$n\in \mathbb{N}$,*and*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$,*for some*$x\in X$,*then*$({x}_{n},x)\in {\bigwedge}_{\alpha}$.

*If we set* $q<\frac{1}{{s}^{2}+s}$, *then* *T* *has a fixed point in* *X*, *that is*, *there exists* $u\in X$ *such that* $u\in Tu$.

*Proof* We can prove this result by using Theorem 3.2 and Remark 2.22. □

**Corollary 3.4** (Theorems 3.2, 3.3 in [10])

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*$s\ge 1$, $\alpha :X\times X\to [0,\mathrm{\infty})$

*be a given mapping and*$T:X\to {P}_{b,cl}(X)$

*be a*

*q*-

*set*-

*valued*

*α*-

*quasi*-

*contraction*,

*that is*,

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

*we have*

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

*Suppose that the following conditions hold*:

- (i)
*T**is**α*-*admissible*(*or*${\alpha}_{\ast}$-*admissible*); - (ii)
*there exist*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$*such that*$({x}_{0},{x}_{1})\in {\bigwedge}_{\alpha}$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$({x}_{n},{x}_{n+1})\in {\bigwedge}_{\alpha}$,*for all*$n\in \mathbb{N},$*and*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$,*for some*$x\in X$,*then*$({x}_{n},x)\in {\bigwedge}_{\alpha}$.

*If* $q<\frac{1}{{s}^{2}+s}$, *then* *T* *has a fixed point in* *X*, *that is*, *there exists* $u\in X$ *such that* $u\in Tu$.

*Proof*We will show that a

*q*-set-valued

*α*-quasi-contraction is a partial

*q*-set-valued quasi-contraction. Assume that $(x,y)\in {\bigwedge}_{\alpha}$ and so $\alpha (x,y)\ge 1$. Since

*T*is a

*q*-set-valued

*α*-quasi-contraction, we get

This implies that *T* is a partial *q*-set-valued quasi-contraction. By Theorem 3.2 (or Theorem 3.3), we get the desired result. □

**Corollary 3.5**

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*$s\ge 1$, $\alpha :X\times X\to [0,\mathrm{\infty})$

*be a given mapping and let*$T:X\to {P}_{b,cl}(X)$

*satisfy*

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

*where*$0\le q<1$

*and*$\u03f5\ge 1$.

*Suppose that the following conditions hold*:

- (i)
*T**is**α*-*admissible*(*or*${\alpha}_{\ast}$-*admissible*); - (ii)
*there exist*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$*such that*$({x}_{0},{x}_{1})\in {\bigwedge}_{\alpha}$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$({x}_{n},{x}_{n+1})\in {\bigwedge}_{\alpha}$,*for all*$n\in \mathbb{N}$,*and*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$,*for some*$x\in X$,*then*$({x}_{n},x)\in {\bigwedge}_{\alpha}$.

*If* $q<\frac{1}{{s}^{2}+s}$, *then* *T* *has a fixed point in* *X*, *that is*, *there exists* $u\in X$ *such that* $u\in Tu$.

*Proof*We will show that

*T*is a partial

*q*-set-valued quasi-contraction. Suppose that $(x,y)\in {\bigwedge}_{\alpha}$ and then $\alpha (x,y)\ge 1$. From (3.12), we get

This implies that *T* is a partial *q*-set-valued quasi-contraction. By Theorem 3.2 (or Theorem 3.3), we get the desired result. □

**Corollary 3.6**

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*$s\ge 1$, $\alpha :X\times X\to [0,\mathrm{\infty})$

*be a given mapping and*$T:X\to {P}_{b,cl}(X)$

*satisfies*

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

*where*$0\le q<1$

*and*$\u03f5>1$.

*Suppose that the following conditions hold*:

- (i)
*T**is**α*-*admissible*(*or*${\alpha}_{\ast}$-*admissible*); - (ii)
*there exist*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$*such that*$({x}_{0},{x}_{1})\in {\bigwedge}_{\alpha}$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*$({x}_{n},{x}_{n+1})\in {\bigwedge}_{\alpha}$,*for all*$n\in \mathbb{N}$,*and*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$,*for some*$x\in X$,*then*$({x}_{n},x)\in {\bigwedge}_{\alpha}$.

*If* $q<\frac{1}{{s}^{2}+s}$, *then* *T* *has a fixed point in* *X*, *that is*, *there exists* $u\in X$ *such that* $u\in Tu$.

*Proof*We will show that

*T*is a partial

*q*-set-valued quasi-contraction. Suppose that $(x,y)\in {\bigwedge}_{\alpha}$ and then $\alpha (x,y)\ge 1$. From (3.13), we get

This implies that *T* is a partial *q*-set-valued quasi-contraction. By Theorem 3.2 (or Theorem 3.3), we get the desired result. □

**Corollary 3.7** (Theorem 2.2 in [8])

*Let* $(X,d)$ *be a complete* *b*-*metric space with coefficient* $s\ge 1$ *and* $T:X\to {P}_{b,cl}(X)$ *be a* *q*-*set*-*valued quasi*-*contraction*. *If* $q<\frac{1}{{s}^{2}+s}$, *then* *T* *has a fixed point in* *X*, *that is*, *there exists* $u\in X$ *such that* $u\in Tu$.

*Proof* Set $\alpha (x,y)=1$, for all $x,y\in X$. By Theorem 3.2 (or Theorem 3.3), we obtain the desired result. □

**Remark 3.8** If we take $s=1$ (it corresponds to the case of metric spaces), then the condition of *q* in Theorem 3.2 becomes $q<\frac{1}{2}$. Therefore, Theorems 3.2 and 3.3 are generalization of several known fixed point results in metric spaces. Also Theorem 3.2 is a generalization of Theorem 3.2 and 3.3 of Sintunavarat *et al.* [10], Theorem 2.2 of Aydi *et al.* [8], main results of Amini-Harandi [9], Daffer and Kaneko [26], Rouhani and Moradi [27], and Singh *et al.* [14].

The following example shows that Theorem 3.2 properly generalizes Theorem 2.2 of Aydi *et al.* [8].

**Example 3.9**Let $X=\mathbb{R}$ and the functional $d:X\times X\to {\mathbb{R}}_{+}$ defined by

*b*-metric space with coefficient $s=2$. Define set-valued mapping $T:X\to {P}_{b,cl}(X)$ by

for all $0\le q<1$. This implies that the contraction condition of Theorem 2.2. of Aydi *et al.* [8] is not true for this case. Therefore, Theorem 2.2 cannot be used to claim the existence of fixed point of *T*.

*T*is a partial

*q*-set-valued quasi-contraction mapping, where $q=\frac{1}{100}$. Assume that

*T*is a partial

*q*-set-valued quasi-contraction mapping. Also we have

It is easy to see that *T* is an *α*-admissible mapping. We find that there exist ${x}_{0}=2$ and ${x}_{1}=0.1\in T{x}_{0}$ for which $({x}_{0},{x}_{1})\in {\bigwedge}_{\alpha}$. Further, for any sequence $\{{x}_{n}\}$ in *X* with ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, for some $x\in X$, and $({x}_{n},{x}_{n+1})\in {\bigwedge}_{\alpha}$, for all $n\in \mathbb{N}$, we see that $({x}_{n},x)\in {\bigwedge}_{\alpha}$, for all $n\in \mathbb{N}$.

Therefore, all hypotheses of Theorem 3.2 are satisfied and so *T* has a fixed point. In this case, *T* have infinitely many fixed points.

## 4 Consequences

### 4.1 Fixed point results of single-valued mappings

In this section, we give the fixed point result for single-valued mappings. Before presenting our results, we introduce the new concept of a partial *q*-single-valued quasi-contraction mapping.

**Definition 4.1**Let $(X,d)$ be a

*b*-metric space and $\alpha :X\times X\to [0,\mathrm{\infty})$ be a mapping. The single-valued mapping $t:X\to X$ is said to be a

*partial*

*q-single-valued quasi-contraction*if

where $0\le q<1$.

Next, we give the fixed point result for partial *q*-single-valued quasi-contraction mapping.

**Theorem 4.2**

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*$s\ge 1$, $\alpha :X\times X\to [0,\mathrm{\infty})$

*be a given mapping and*$t:X\to X$

*be a partial*

*q*-

*single*-

*valued quasi*-

*contraction*.

*Suppose that the following conditions hold*:

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

*If* $q<\frac{1}{{s}^{2}+s}$, *then* *t* *has a fixed point in* *X*, *that is*, *there exists* $u\in X$ *such that* $u=tu$.

*Proof* It follows by applying Theorem 3.2 or Theorem 3.3. □

**Remark 4.3** Theorem 4.2 is an extension of Corollary 3.8 of Sintunavarat *et al.* [10], Corollary 2.4 of Aydi *et al.* [8], and the result of Ćirić [28].

**Example 4.4**Let $X=\mathbb{R}$ and the functional $d:X\times X\to {\mathbb{R}}_{+}$ defined by

*b*-metric space with coefficient $s=2$. Define single-valued mapping $t:X\to X$ by

for all $0\le q<1$. This implies that the contraction condition of Corollary 2.4 of Aydi *et al.* [8] is not true for this case. Therefore, Corollary 2.4 of Aydi *et al.* [8] cannot be used to claim the existence of fixed point of *t*.

*t*is a partial

*q*-single-valued quasi-contraction mapping, where $q=\frac{1}{9}$. Assume that $(x,y)\in {\bigwedge}_{\alpha}=\{(x,y)\in X\times X:\alpha (x,y)\ge 1\}=[0,1]\times [0,1]$. We obtain

*t*is a partial

*q*-single-valued quasi-contraction mapping. Also we have

It is easy to see that *t* is an *α*-admissible mapping.

We find that there exists ${x}_{0}=0.3$ such that $({x}_{0},t{x}_{0})=(0.3,0.1)\in {\bigwedge}_{\alpha}$. Further, for any sequence $\{{x}_{n}\}$ in *X* with ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, for some $x\in X$, and $({x}_{n},{x}_{n+1})\in {\bigwedge}_{\alpha}$, for all $n\in \mathbb{N}$, we obtain $({x}_{n},x)\in {\bigwedge}_{\alpha}$, for all $n\in \mathbb{N},$ since $[0,1]$ is closed.

Therefore, all hypotheses of Theorem 4.2 are satisfied and so *t* has a fixed point, that is, a point $0\in X$.

### 4.2 Fixed point results on *b*-metric space endowed with an arbitrary binary relation

In this section, we give the fixed point results on a *b*-metric space endowed with an arbitrary binary relation. Before presenting our results, we give the following definitions.

**Definition 4.5** Let $(X,d)$ be a *b*-metric space and ℛ be a binary relation over *X*. We say that $T:X\to {P}_{b,cl}(X)$ is a weakly preserving mapping if for each $x\in X$ and $y\in Tx$ with $x\mathcal{R}y$, we have $y\mathcal{R}z$, for all $z\in Ty$.

**Definition 4.6**Let $(X,d)$ be a

*b*-metric space and ℛ be a binary relation over

*X*. The set-valued mapping $T:X\to {P}_{b,cl}(X)$ is said to be a

*q*-set-valued quasi-contraction with respect to ℛ if, for all $x,y\in X$, we have

where $0\le q<1$.

**Theorem 4.7**

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*$s\ge 1$, ℛ

*be a binary relation over*

*X*,

*and*$T:X\to {P}_{b,cl}(X)$

*be a*

*q*-

*set*-

*valued quasi*-

*contraction with respect to*ℛ.

*Suppose that the following conditions hold*:

- (i)
*T**is a weakly preserving mapping*; - (ii)
*there exist*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$*such that*${x}_{0}\mathcal{R}{x}_{1}$; - (iii)
*if*$\{{x}_{n}\}$*is a sequence in**X**such that*${x}_{n}\mathcal{R}{x}_{n+1}$,*for all*$n\in \mathbb{N}$,*and*${x}_{n}\to x$*as*$n\to \mathrm{\infty}$,*for some*$x\in X$,*then*${x}_{n}\mathcal{R}x$.

*If* $q<\frac{1}{{s}^{2}+s}$, *then* *T* *has a fixed point in* *X*, *that is*, *there exists* $u\in X$ *such that* $u\in Tu$.

*Proof*Consider the mapping $\alpha :X\times X\to [0,\mathrm{\infty})$ defined by

*T*being a preserving mapping that

*T*is an

*α*-admissible mapping. Since

*T*is a

*q*-set-valued quasi-contraction with respect to ℛ, we have, for all $x,y\in X$,

This implies that *T* is a partial *q*-set-valued quasi-contraction mapping. Now all the hypotheses of Theorem 3.2 are satisfied and so the existence of the fixed point of *T* follows from Theorem 3.2. □

Next, we give some special case of Theorem 4.7 in partially ordered *b*-metric spaces. Before we study the next results, we give the following definitions.

**Definition 4.8** Let *X* be a nonempty set. Then $(X,d,\u2aaf)$ is called a partially ordered *b*-metric space if $(X,d)$ is a *b*-metric space and $(X,\u2aaf)$ is a partially ordered space.

**Definition 4.9** Let $(X,d,\u2aaf)$ be a partially ordered *b*-metric space. We say that $T:X\to {P}_{b,cl}(X)$ is a weakly preserving mapping with ⪯ if for each $x\in X$ and $y\in Tx$ with $x\u2aafy$, we have $y\u2aafz$, for all $z\in Ty$.

**Definition 4.10**Let $(X,d,\u2aaf)$ be a partially ordered

*b*-metric space. The set-valued mapping $T:X\to {P}_{b,cl}(X)$ is said to be a

*q*-set-valued quasi-contraction with respect to ⪯ if, for all $x,y\in X$, we have

where $0\le q<1$.

**Corollary 4.11**

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

*be a complete partially ordered*

*b*-

*metric space with coefficient*$s\ge 1$

*and*$T:X\to {P}_{b,cl}(X)$

*be a*

*q*-

*set*-

*valued quasi*-

*contraction with respect to*⪯.

*Suppose that the following conditions hold*:

- (i)
*T**is a weakly preserving mapping with*⪯; - (ii)
*there exist*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$*such that*${x}_{0}\u2aaf{x}_{1}$; - (iii)
*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*$n\to \mathrm{\infty}$,*for some*$x\in X$,*then*${x}_{n}\u2aafx$.

*If we set* $q<\frac{1}{{s}^{2}+s}$, *then* *T* *has a fixed point in* *X*, *that is*, *there exists* $u\in X$ *such that* $u\in Tu$.

*Proof* The result follows from Theorem 4.7 by considering the binary relation ⪯. □

### 4.3 Fixed point results on *b*-metric spaces endowed with a graph

Throughout this section, let $(X,d)$ be a *b*-metric space. A set $\{(x,x):x\in X\}$ is called a diagonal of the Cartesian product $X\times X$ and is denoted by Δ. 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.*, $\mathrm{\Delta}\subseteq E(G)$. We assume that *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 by assigning to each edge the distance between its vertices.

In this section, we give the fixed point results for set-valued mappings in a *b*-metric space endowed with a graph. Before presenting our results, we will introduce new definitions in a *b*-metric space endowed with a graph.

**Definition 4.12** Let $(X,d)$ be a *b*-metric space endowed with a graph *G* and $T:X\to {P}_{b,cl}(X)$ be set-valued mapping. We say that *T* weakly preserves the edges of *G* if for each $x\in X$ and $y\in Tx$ with $(x,y)\in E(G)$ implies $(y,z)\in E(G)$, for all $z\in Ty$.

**Definition 4.13**Let $(X,d)$ be a

*b*-metric space endowed with a graph

*G*. A set-valued mapping $T:X\to {P}_{b,cl}(X)$ is said to be a

*q*-

*G*-set-valued quasi-contraction if, for all $x,y\in X$, we have

where $0\le q<1$.

**Example 4.14** Let *X* be a nonempty set. Any mapping $T:X\to {P}_{b,cl}(X)$ defined by $Tx=\{a\}$, where $a\in X$, is a *q*-*G*-set-valued quasi-contraction for any graph *G* with $V(G)=X$.

**Example 4.15** Let *X* be a nonempty set. Any mapping $T:X\to {P}_{b,cl}(X)$ is trivially a *q*-*G*-set-valued quasi-contraction, where $G=(V(G),E(G))=(X,\mathrm{\Delta})$.

**Definition 4.16** Let $(X,d)$ be a *b*-metric space endowed with a graph *G*. We say that *X* has *G*-regular property if given $x\in X$ and sequence $\{{x}_{n}\}$ in *X* such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$ and $({x}_{n},{x}_{n+1})\in E(G)$, for all $n\in \mathbb{N}$, then $({x}_{n},u)\in E(G)$, for all $n\in \mathbb{N}$.

Here, we give a fixed point result for set-valued mappings in a *b*-metric space endowed with a graph.

**Theorem 4.17**

*Let*$(X,d)$

*be a complete*

*b*-

*metric space with coefficient*$s\ge 1$

*and endowed with a graph*

*G*

*and let*$T:X\to {P}_{b,cl}(X)$

*be a*

*q*-

*G*-

*set*-

*valued quasi*-

*contraction*.

*Suppose that the following conditions hold*:

- (i)
*T**weakly preserves edges of**G*; - (ii)
*there exist*${x}_{0}\in X$*and*${x}_{1}\in T{x}_{0}$*such that*$({x}_{0},{x}_{1})\in E(G)$; - (iii)
*X**has**G*-*regular property*.

*If* $q<\frac{1}{{s}^{2}+s}$, *then* *T* *has a fixed point in* *X*, *that is*, *there exists* $u\in X$ *such that* $u\in Tu$.

*Proof*Consider the mapping $\alpha :X\times X\to [0,\mathrm{\infty})$ defined by

*T*is a

*q*-

*G*-set-valued quasi-contraction, we have, for all $x,y\in X$,

This implies that *T* is a partial *q*-set-valued quasi-contraction.

By construction of *α* and condition (i), we find that *T* is *α*-admissible. From condition (ii) and the construction of *α*, we get $\alpha ({x}_{0},{x}_{1})\ge 1$ and thus $({x}_{0},{x}_{1})\in {\bigwedge}_{\alpha}$. Using *G*-regular property of *X*, the result is obtained that the condition (iii) in Theorem 3.2 holds. Now all the hypotheses of Theorem 3.2 are satisfied and so the existence of the fixed point of *T* follows from Theorem 3.2. □

## 5 Existence of a solution for an integral equation

where $b>a\ge 0$, $x\in C[a,b]$ (the set of continuous real functions defined on $[a,b]\subseteq \mathbb{R}$), $\varphi :[a,b]\to \mathbb{R}$, and $K:[a,b]\times [a,b]\times \mathbb{R}\to \mathbb{R}$ are given mappings.

**Theorem 5.1** *Suppose that the following hypotheses hold*:

(I_{1}) $K:[a,b]\times [a,b]\times \mathbb{R}\to \mathbb{R}$ *is continuous*;

_{2})

*there exists*$p\ge 1$

*satisfies the following condition for each*$r,c\in [a,b]$

*and*$x,y\in X$

*with*$x(w)\le y(w)$,

*for all*$w\in [a,b]$:

*where*$\xi :[a,b]\times [a,b]\to [0,\mathrm{\infty})$

*is a continuous function satisfying*

(I_{3}) *there exists* ${x}_{0}\in X$ *such that* ${x}_{0}(c)\le (t{x}_{0})(c)$, *for all* $c\in [a,b]$.

*Then the integral equation* (5.1) *has a solution* $x\in X$.

*Proof*Let $X=C[a,b]$ and let $t:X\to X$ be a mapping defined by

*X*with the

*b*-metric $d:X\times X\to {\mathbb{R}}_{+}$ given by

for all $x,y\in X$, is a complete *b*-metric space with coefficient $s={2}^{p-1}$.

It is easy to see that *t* is an *α*-admissible mapping. From (I_{3}), we have $({x}_{0},t{x}_{0})\in {\bigwedge}_{\alpha}$. Also we find that condition (iii) in Theorem 4.2 holds (see [29]).

*t*is a partial

*q*-single-valued quasi-contraction mapping with $q=\frac{1}{{2}^{p-1}+1}<\frac{1}{{2}^{p-1}({2}^{p-1}+1)}=\frac{1}{{s}^{2}+s}$. Let $1\le {p}^{\prime}<\mathrm{\infty}$ with $\frac{1}{p}+\frac{1}{{p}^{\prime}}=1$. Now, let $x,y\in X$ be such that $(x,y)\in {\bigwedge}_{\alpha}$, that is, $x(c)\le y(c)$, for all $c\in [a,b]$. From (I

_{1}), (I

_{2}), and the Hölder inequality, for each $s\in [a,b]$ we have

Therefore, by using Theorem 4.2, we see that *t* has a fixed point, that is, there exists $x\in X$ such that *x* is a fixed point of *t*. This implies that *x* is a solution for (5.1) because the existence of a solution of (5.1) is equivalent to the existence of a fixed point of *t*. This completes the proof. □

## Declarations

### Acknowledgements

The second author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript.

## Authors’ Affiliations

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