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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
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 mappings
- binary relations
- fixed 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.
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.
Consequently, condition (B_{3}) is also satisfied and thus ρ is a b-metric on X.
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.
is a b-metric space with constant $s={2}^{\frac{1}{p}}$.
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])
- (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 $m,n\to \mathrm{\infty}$.
- (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])
Therefore, $(X,d)$ is a b-metric space on X with coefficient $s=3$.
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])
Definition 2.13 ([4])
- (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 $\delta (Y):=sup\{d(a,b)\mid a,b\in Y\}<\mathrm{\infty}$.
Next, we give the concept of generalized functionals on a b-metric space $(X,d)$.
- (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$H(A,B)=\{\begin{array}{ll}max\{\rho (A,B),\rho (B,A)\},& A\ne \mathrm{\varnothing}\ne B,\\ 0,& A=\mathrm{\varnothing},\\ +\mathrm{\infty},& \text{otherwise}.\end{array}$
- (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])
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])
Lemma 2.19 ([14])
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])
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.
- (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.
where $0\le q<1$.
Next, we give the main result in this paper.
- (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 $u\in Tu$.
for all $x,y\in X$.
It follows from $q<\frac{1}{{s}^{2}+s}$ that $\epsilon >0$ and $0<\beta <\frac{1}{{s}^{2}+s}$.
where $\gamma :=\frac{\beta s}{1-\beta s}$.
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. □
- (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])
- (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 $u\in Tu$.
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. □
- (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 $u\in Tu$.
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. □
- (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 $u\in Tu$.
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].
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.
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.
where $0\le q<1$.
Next, we give the fixed point result for partial q-single-valued quasi-contraction mapping.
- (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].
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.
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 $z\in Ty$.
where $0\le q<1$.
- (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 $u\in Tu$.
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 $z\in Ty$.
where $0\le q<1$.
- (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 $z\in Ty$.
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.
- (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 $u\in Tu$.
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;
(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 $x\in X$.
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]).
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
References
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