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The existence of fixed point theorems for partial qsetvalued quasicontractions in bmetric spaces and related results
Fixed Point Theory and Applicationsvolume 2014, Article number: 226 (2014)
Abstract
In this paper, we present a new type of setvalued mappings called partial qsetvalued quasicontraction mappings and give results as regards fixed points for such mappings in bmetric 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 singlevalued mapping, fixed point results for setvalued mapping in bmetric space endowed with an arbitrary binary relation, and fixed point results in a bmetric 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.
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 bmetric spaces and also presented the fixed point theorem for contraction mappings in bmetric spaces, that is, we have a generalization of the Banach contraction principle in metric spaces. Afterward, many mathematicians studied fixed point theorems for singlevalued and setvalued mappings in bmetric spaces (see [2–7] and references therein).
In 2012, Aydi et al. [8] extended the concept of qsetvalued quasicontraction mappings in metric spaces due to AminiHarandi [9] to bmetric spaces. They also established the fixed point results for qsetvalued quasicontraction mappings in bmetric spaces. Recently, Sintunavarat et al. [10] introduced some setvalued mappings called qsetvalued αquasicontraction mappings and obtained fixed point results for such mappings in bmetric spaces which are generalization of the results of Aydi et al. [8], AminiHarandi [9] and many works in the literature.
Inspired and motivated by several results in the literature, we introduce the class of partial qsetvalued quasicontraction mappings which is the wider class of many classes in this field. As regards this class, we study and obtain fixed point results in bmetric spaces. These results extend, unify and generalize several wellknown comparable results in the existing literature. As an application of our results, we prove the fixed point theorems for a singlevalued mapping and give an example to show the generality of our result. We also study the fixed point results in a bmetric 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 bmetric 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 bmetric 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 bmetric space with coefficient s.
Remark 2.2 The result is obtained that any metric space is a bmetric space with $s=1$. Thus the class of bmetric spaces is larger than the class of metric spaces.
Some examples of bmetric spaces are given by Berinde [11], Czerwik [6], Heinonen [12]. Some wellknown examples of a bmetric which show that the bmetric space is a real generalization of metric space are the following.
Example 2.3 The set of real numbers together with the functional $d:\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}_{+}$,
for all $x,y\in \mathbb{R}$, is a bmetric 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 bmetric with $s={2}^{p1}$. 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:
that is,
holds. Therefore, for each $x,y,z\in X$, we get
Consequently, condition (B_{3}) is also satisfied and thus ρ is a bmetric on X.
Example 2.5 The set ${l}_{p}(\mathbb{R})$ with $0<p<1$, where
together with the functional $d:{l}_{p}(\mathbb{R})\times {l}_{p}(\mathbb{R})\to {\mathbb{R}}_{+}$,
for each $x=\{{x}_{n}\},y=\{{y}_{n}\}\in {l}_{p}(\mathbb{R})$, is a bmetric 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 bmetric 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
and
where 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 bmetric 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 bmetric space.
Definition 2.8 ([4])
Let $(X,d)$ be a bmetric 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 bmetric 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 bmetric $d:X\times X\to {\mathbb{R}}_{+}$ for coefficient $s>1$ is not jointly continuous in all its variables.
The following example is an example of a bmetric which is not continuous.
Example 2.10 (see [13])
Let $X=\mathbb{N}\cup \{\mathrm{\infty}\}$ and a functional $d:X\times X\to {\mathbb{R}}_{+}$ be defined by
It is easy to see that conditions (B_{1}) and (B_{2}) are satisfied. Also, for each $x,y,z\in X$, we have
Therefore, $(X,d)$ is a bmetric space on X with coefficient $s=3$.
Next, we show that 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 bmetric space $(X,d)$ is complete if every Cauchy sequence in X converges.
Definition 2.12 ([4])
Let Y be a nonempty subset of a bmetric 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])
Let $(X,d)$ be a bmetric 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 $\delta (Y):=sup\{d(a,b)\mid a,b\in Y\}<\mathrm{\infty}$.
Throughout this paper, we use the following notations of collection of subsets of a bmetric space $(X,d)$:
Next, we give the concept of generalized functionals on a bmetric space $(X,d)$.
Definition 2.14 Let $(X,d)$ be a bmetric 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 PompeiuHausdorff 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 bmetric space $(X,d)$, the following assertions hold:

(1)
$({P}_{cp}(X),H)$ is a complete bmetric space provided $(X,d)$ is a complete bmetric 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 bmetric 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 bmetric 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 bmetric space. For $A\in {P}_{b,cl}(X)$ and $x\in X$, we have
Lemma 2.19 ([14])
Let $(X,d)$ be a bmetric 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 αadmissible mapping as follows.
Definition 2.20 ([15])
Let X be a nonempty set, $t:X\to X$ and $\alpha :X\times X\to [0,\mathrm{\infty})$. We say that t is αadmissible if
They proved the fixed point results for singlevalued 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 αadmissibility for setvalued mappings as follows.
Let 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 αadmissible 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 αadmissible mapping.
In recent investigations, the fixed point results for singlevalued and setvalued mappings via the concepts of being αadmissible and ${\alpha}_{\ast}$admissible occupies a prominent place in many aspects (see [18–25] and references therein).
3 Fixed point theorems for partial qsetvalued quasicontraction mappings
In this section, we introduce the partial qsetvalued quasicontraction mapping and obtain the theorem of the existence of a fixed point for such a mapping in bmetric spaces.
Throughout this paper, for the nonempty set 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 bmetric space and $\alpha :X\times X\to [0,\mathrm{\infty})$ be a given mapping. The setvalued mapping $T:X\to {P}_{b,cl}(X)$ is said to be a partial qsetvalued quasicontraction 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 bmetric 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 qsetvalued quasicontraction. Suppose that the following conditions hold:

(i)
T is αadmissible;

(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
if and only if $x=y$ is a fixed point of T. Therefore, we suppose that
for all $x,y\in X$.
Now, we will set
It follows from $q<\frac{1}{{s}^{2}+s}$ that $\epsilon >0$ and $0<\beta <\frac{1}{{s}^{2}+s}$.
Starting from ${x}_{0}$ and ${x}_{1}\in T{x}_{0}$ in (ii), by Lemma 2.17, there exists ${x}_{2}\in T{x}_{1}$ such that
It follows from $({x}_{0},{x}_{1})\in {\bigwedge}_{\alpha}$ that
From (3.2) and (3.3), we get
Since T is αadmissible, ${x}_{0}\in X$, 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
Since T is a partial qsetvalued quasicontraction and $({x}_{1},{x}_{2})\in {\bigwedge}_{\alpha}$, we obtain
From (3.4) and (3.5), we have
By induction, we can construct a sequence $\{{x}_{n}\}$ in X such that, for each $n\in \mathbb{N}$, we have
and
If there exists $\stackrel{\u02c6}{n}\in \mathbb{N}$ such that ${x}_{\stackrel{\u02c6}{n}1}={x}_{\stackrel{\u02c6}{n}}$, then ${x}_{\stackrel{\u02c6}{n}}\in T{x}_{\stackrel{\u02c6}{n}}$ and then the proof is complete. For the rest, we will assume that ${x}_{n1}\ne {x}_{n}$, that is, $d({x}_{n1},{x}_{n})>0$, for all $n\in \mathbb{N}$. Now we obtain, for all $n\in \mathbb{N}$,
and hence
where $\gamma :=\frac{\beta s}{1\beta s}$.
Since $s\ge 1$, $\beta =\frac{1}{2}(\frac{1}{{s}^{2}+s}+q)$, and $q<\frac{1}{{s}^{2}+s}$, we get
From (3.7), (3.8), and Lemma 2.19, we see that $\{{x}_{n}\}$ is a Cauchy sequence in X. By the completeness of X, there exists $u\in X$ such that
Next, we will prove that $d(u,Tu)=0$. By the condition (iii), we have $({x}_{n},u)\in {\bigwedge}_{\alpha}$, for all $n\in \mathbb{N}$. From Lemma 2.16 and (3.1), for each $n\in \mathbb{N}$, we get
Letting $n\to \mathrm{\infty}$ in the above inequality, we have
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 bmetric 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 qsetvalued quasicontraction. 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 bmetric 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 qsetvalued αquasicontraction, that is, for all $x,y\in X$, we have
where $0\le q<1$. Suppose that the following conditions hold:

(i)
T is αadmissible (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 qsetvalued αquasicontraction is a partial qsetvalued quasicontraction. Assume that $(x,y)\in {\bigwedge}_{\alpha}$ and so $\alpha (x,y)\ge 1$. Since T is a qsetvalued αquasicontraction, we get
This implies that T is a partial qsetvalued quasicontraction. By Theorem 3.2 (or Theorem 3.3), we get the desired result. □
Corollary 3.5 Let $(X,d)$ be a complete bmetric 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 αadmissible (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 qsetvalued quasicontraction. Suppose that $(x,y)\in {\bigwedge}_{\alpha}$ and then $\alpha (x,y)\ge 1$. From (3.12), we get
that is,
This implies that T is a partial qsetvalued quasicontraction. By Theorem 3.2 (or Theorem 3.3), we get the desired result. □
Corollary 3.6 Let $(X,d)$ be a complete bmetric 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 αadmissible (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 qsetvalued quasicontraction. Suppose that $(x,y)\in {\bigwedge}_{\alpha}$ and then $\alpha (x,y)\ge 1$. From (3.13), we get
It follows from $\u03f5>1$ that
This implies that T is a partial qsetvalued quasicontraction. 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 bmetric space with coefficient $s\ge 1$ and $T:X\to {P}_{b,cl}(X)$ be a qsetvalued quasicontraction. 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 AminiHarandi [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
for all $x,y\in X$. Clearly, $(X,d)$ is a complete bmetric space with coefficient $s=2$. Define setvalued mapping $T:X\to {P}_{b,cl}(X)$ by
and $\alpha :X\times X\to [0,\mathrm{\infty})$ by
We obtain
and
Therefore,
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.
Next, we show that Theorem 3.2 can be applied for this case. First of all, we show that T is a partial qsetvalued quasicontraction mapping, where $q=\frac{1}{100}$. Assume that
Then we have
This shows that T is a partial qsetvalued quasicontraction mapping. Also we have
It is easy to see that T is an αadmissible 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 singlevalued mappings
In this section, we give the fixed point result for singlevalued mappings. Before presenting our results, we introduce the new concept of a partial qsinglevalued quasicontraction mapping.
Definition 4.1 Let $(X,d)$ be a bmetric space and $\alpha :X\times X\to [0,\mathrm{\infty})$ be a mapping. The singlevalued mapping $t:X\to X$ is said to be a partial qsinglevalued quasicontraction if
where $0\le q<1$.
Next, we give the fixed point result for partial qsinglevalued quasicontraction mapping.
Theorem 4.2 Let $(X,d)$ be a complete bmetric 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 qsinglevalued quasicontraction. Suppose that the following conditions hold:

(i)
t is αadmissible;

(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
for all $x,y\in X$. Clearly, $(X,d)$ is a complete bmetric space with coefficient $s=2$. Define singlevalued mapping $t:X\to X$ by
and $\alpha :X\times X\to [0,\mathrm{\infty})$ by
We obtain
and
Therefore,
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.
Next, we show that Theorem 4.2 can be applying for this case. First of all, we show that t is a partial qsinglevalued quasicontraction 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
This shows that t is a partial qsinglevalued quasicontraction mapping. Also we have
It is easy to see that t is an αadmissible 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 bmetric space endowed with an arbitrary binary relation
In this section, we give the fixed point results on a bmetric space endowed with an arbitrary binary relation. Before presenting our results, we give the following definitions.
Definition 4.5 Let $(X,d)$ be a bmetric 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 bmetric space and ℛ be a binary relation over X. The setvalued mapping $T:X\to {P}_{b,cl}(X)$ is said to be a qsetvalued quasicontraction 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 bmetric space with coefficient $s\ge 1$, ℛ be a binary relation over X, and $T:X\to {P}_{b,cl}(X)$ be a qsetvalued quasicontraction 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
From condition (ii), we get $\alpha ({x}_{0},{x}_{1})\ge 1$ and so $({x}_{0},{x}_{1})\in {\bigwedge}_{\alpha}$. It follows from T being a preserving mapping that T is an αadmissible mapping. Since T is a qsetvalued quasicontraction with respect to ℛ, we have, for all $x,y\in X$,
This implies that T is a partial qsetvalued quasicontraction 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 bmetric 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 bmetric space if $(X,d)$ is a bmetric space and $(X,\u2aaf)$ is a partially ordered space.
Definition 4.9 Let $(X,d,\u2aaf)$ be a partially ordered bmetric 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 bmetric space. The setvalued mapping $T:X\to {P}_{b,cl}(X)$ is said to be a qsetvalued quasicontraction 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 bmetric space with coefficient $s\ge 1$ and $T:X\to {P}_{b,cl}(X)$ be a qsetvalued quasicontraction 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 bmetric spaces endowed with a graph
Throughout this section, let $(X,d)$ be a bmetric 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 setvalued mappings in a bmetric space endowed with a graph. Before presenting our results, we will introduce new definitions in a bmetric space endowed with a graph.
Definition 4.12 Let $(X,d)$ be a bmetric space endowed with a graph G and $T:X\to {P}_{b,cl}(X)$ be setvalued 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 bmetric space endowed with a graph G. A setvalued mapping $T:X\to {P}_{b,cl}(X)$ is said to be a qGsetvalued quasicontraction 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 qGsetvalued quasicontraction 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 qGsetvalued quasicontraction, where $G=(V(G),E(G))=(X,\mathrm{\Delta})$.
Definition 4.16 Let $(X,d)$ be a bmetric space endowed with a graph G. We say that X has Gregular 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 setvalued mappings in a bmetric space endowed with a graph.
Theorem 4.17 Let $(X,d)$ be a complete bmetric space with coefficient $s\ge 1$ and endowed with a graph G and let $T:X\to {P}_{b,cl}(X)$ be a qGsetvalued quasicontraction. 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 Gregular 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
Since T is a qGsetvalued quasicontraction, we have, for all $x,y\in X$,
This implies that T is a partial qsetvalued quasicontraction.
By construction of α and condition (i), we find that T is αadmissible. 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 Gregular 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
In this section, we prove the existence theorem for a solution of the following integral equation by using Theorem 4.2:
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;
(I_{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
for all $x\in X$ and $c\in [a,b]$. Clearly, X with the bmetric $d:X\times X\to {\mathbb{R}}_{+}$ given by
for all $x,y\in X$, is a complete bmetric space with coefficient $s={2}^{p1}$.
Define a mapping $\alpha :X\times X\to [0,\mathrm{\infty})$ by
It is easy to see that t is an αadmissible 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]).
Next, we show that t is a partial qsinglevalued quasicontraction mapping with $q=\frac{1}{{2}^{p1}+1}<\frac{1}{{2}^{p1}({2}^{p1}+1)}=\frac{1}{{s}^{2}+s}$. 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
This shows that
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. □
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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.
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Keywords
 αadmissible mappings
 binary relations
 fixed points
 bmetric spaces
 qsetvalued αquasicontraction mappings