 Research
 Open Access
Some fixed point theorems in bmetriclike spaces
 Chunfang Chen^{1},
 Jian Dong^{1} and
 Chuanxi Zhu^{1}Email author
https://doi.org/10.1186/s1366301503693
© Chen et al. 2015
 Received: 29 October 2014
 Accepted: 26 June 2015
 Published: 21 July 2015
Abstract
In this work, some fixed point and common fixed point theorems are investigated in bmetriclike spaces. Some of our results generalize related results in the literature. Also, some examples and an application to integral equation are given to support our main results.
Keywords
 bmetriclike spaces
 common fixed point
 fixed point
MSC
 47H10
 54H25
1 Introduction and preliminaries
There exist many generalizations of the concept of metric spaces in the literature. In [1, 2], Matthews introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. A lot of fixed point theorems were investigated in partial spaces (see, e.g., [3–11] and references therein). The notions of metriclike spaces [12] and bmetric spaces [13–16] were introduced in the literature, which are generalizations of metric spaces. Recently, the concept of bmetriclike spaces which is a generalization of metriclike spaces and bmetric spaces and partial metric spaces was introduced in [17]. Recently, Hussain et al. [18] discussed topological structure of bmetriclike spaces and proved some fixed point results in bmetriclike spaces.
Definition 1.1
[17]
 (D_{1}):

if \(D(x,y)=0 \) then \(x=y\);
 (D_{2}):

\(D(x,y)=D(y,x)\);
 (D_{3}):

\(D(x,z)\leq s(D(x,y)+D(y,z))\).
Example 1.1
In [17], some concepts in bmetriclike spaces were introduced as follows.
Each bmetriclike D on X generalizes a topology \(\tau_{D}\) on X whose base is the family of open Dballs \(B_{D}(x,\varepsilon)=\{y\in X:D(x,y)D(x,x)<\varepsilon\}\), for all \(x\in X\) and \(\varepsilon>0\).
A sequence \(\{x_{n}\}\) in the bmetriclike space \((X,D)\) converges to a point \(x\in X\) if and only if \(D(x,x)=\lim_{n\rightarrow+\infty}D(x,x_{n})\).
A sequence \(\{x_{n}\}\) in the bmetriclike space \((X,D)\) is called a Cauchy sequence if there exists \(\lim_{n,m\rightarrow+\infty}D(x_{m},x_{n})\) (and it is finite).
A bmetriclike space is called complete if every Cauchy sequence \(\{ x_{n}\}\) in X converges with respect to \(\tau_{D}\) to a point \(x\in X\) such that \(\lim_{n\rightarrow+\infty}D(x,x_{n})=D(x,x)=\lim_{n,m\rightarrow+\infty}D(x_{m},x_{n})\).
Remark 1.1
In Example 1.1, let \(x_{n}=2\) for each \(n=1,2,\ldots\) , then it is clear that \(\lim_{n\rightarrow+\infty}D(x_{n},2)=D(2,2)\) and \(\lim_{n\rightarrow+\infty}D(x_{n},1)=D(1,1)\), hence, in bmetriclike spaces, the limit of a convergent sequence is not necessarily unique.
Remark 1.2
It should be noted that in general, a bmetriclike function \(D(x,y)\) need not be jointly continuous in both variables. The following example illustrates this fact.
Example 1.2
Definition 1.2
Suppose that \((X,D)\) is a bmetriclike space. A mapping \(T:X\rightarrow X\) is said to be continuous at \(x\in X\), if for every \(\varepsilon>0\) there exists \(\delta>0\) such that \(T(B_{D}(x,\delta ))\subseteq B_{D}(Tx,\varepsilon)\). We say that T is continuous on X if T is continuous at all \(x\in X\).
In this paper, we investigate some new fixed point and common fixed point theorems in bmetriclike spaces. Some of our results generalize and improve related results in the literature. Some examples and an application are presented to support our main results.
2 Main results
In this section, we begin with the following definitions and lemma which will be needed in the sequel.
Definition 2.1
[19]
Let f and g be two selfmappings on a set X. If \(\omega=fx=gx\) for some x in X, then x is called a coincidence point of f and g, where ω is called a point of coincidence of f and g.
Definition 2.2
[19]
Let f and g be two selfmappings defined on a set X. Then f and g are said to be weakly compatible if they commute at every coincidence point, i.e., if \(fx=gx\) for some \(x\in X\), then \(fgx=gfx\).
Lemma 2.1
[17]
Then \(\lim_{m,n\rightarrow+\infty}D(y_{m},y_{n})=0\).
 (1)
ϕ is continuous and nondecreasing;
 (2)
\(\phi(t)=0\) if and only if \(t=0\).
Now we prove our main results.
Theorem 2.1
Proof
We claim that if \(D(x,x_{N_{0}})\leq\varepsilon\) for \(N_{0}>N\), then \(D(Tx,x_{N_{0}})\leq\varepsilon\). For this, we distinguish two cases.
In Theorem 2.1, taking \(\varphi(t)=\frac{t}{s}\lambda t\) with \(0<\lambda<\frac{1}{s}\), we can get the following corollary.
Corollary 2.1
Theorem 2.2
Proof
Corollary 2.2
Proof
Letting \(a_{i}=0\) (\(i=2,3,4\)) and \(a_{1}=k\), we find that T has a fixed point from Theorem 2.2. Suppose that u and v are fixed points of T, then we get \(D(u,v)=0\) (otherwise \(D(u,v)=D(Tu,Tv)\geq kD(u,v)>D(u,v)\), which is a contradiction), hence \(u=v\), therefore T has a unique fixed point. □
Lemma 2.2
[20]
Let X be a nonempty set and \(T:X\rightarrow X\) a function. Then there exists a subset \(E\subseteq X\) such that \(T(E)=T(X)\) and \(T:E\rightarrow X\) is onetoone.
Corollary 2.3
Proof
By Lemma 2.2, there exists \(E\subseteq X\) such that \(T(E)=T(X)\) and \(T:E\rightarrow X\) is onetoone. Now, we define a mapping \(h:T(E)\rightarrow T(E)\) by \(h(Tx)=Fx\). Since T is onetoone on E, h is well defined. Note that \(D(h(Tx),h(Ty))\geq kD(Tx,Ty)\) for all \(Tx,Ty\in T(E)\). Since \(T(E)=T(X)\) is complete, by using Corollary 2.2, there exists a unique \(x_{0}\in X\) such that \(h(Tx_{0})=Tx_{0}\), hence \(Fx_{0}=Tx_{0}\), which means that F and T have a unique point of coincidence in X. Let \(Fx_{0}=Tx_{0}=z\), since F and T are weakly compatible, \(Fz=Tz\), which together with the uniqueness of point of coincidence implies that \(Fz=Tz=z\), therefore, z is the unique common fixed point of F and T. □
Now, we introduce some examples to illustrate the validity of our main results.
Example 2.1
 (1)
\((X,D)\) is a complete bmetriclike space with the constant \(s=\frac{8}{5}\).
 (2)
For all \(x,y\in X\), we have \(D(Tx,Ty)\leq\frac{D(x,y)}{s}\varphi (D(x,y))\).
Proof
Example 2.2
3 Existence of a solution for an integral equation
Let \(f(x(t))=\int_{0}^{T}K(t,r,x(r))\, dr\) for all \(x\in X\) and for all \(t\in[0,T]\). Then the existence of a solution to (3.1) is equivalent to the existence of a fixed point of f. Now, we prove the following result.
Theorem 3.1
 (i)
\(K:[0,T]\times[0,T]\times R\rightarrow R\) is continuous;
 (ii)for all \(t,r\in[0,T]\), there exists a continuous \(\xi :[0,T]\times[0,T]\rightarrow R\) such thatand$$ \bigl\vert K\bigl(t,r,x(r)\bigr)\bigr\vert +\bigl\vert K \bigl(t,r,y(r)\bigr)\bigr\vert < \lambda^{\frac{1}{p}}\xi(t,r) \bigl(\bigl\vert x(r)\bigr\vert +\bigl\vert y(r)\bigr\vert \bigr) $$(3.2)where \(0<\lambda<\frac{1}{s}\).$$ \sup_{t\in[0,T]}\int_{0}^{T} \xi(t,r)\, dr \leq1, $$(3.3)
Proof
Now, all the conditions of Corollary 2.1 hold and f has a unique fixed point \(x\in X\), which means that x is the unique solution for the integral equation (3.1). □
Declarations
Acknowledgements
The authors are thankful to the referees for their valuable comments and suggestions to improve this paper. The research was supported by the National Natural Science Foundation of China (11361042, 11071108, 11461045) and supported by the Provincial Natural Science Foundation of Jiangxi, China (20114BAB201007, 20132BAB201001, 20142BAB201007, 20142BAB211004, 20142BAB211016) and the Science and Technology Project of Educational Commission of Jiangxi Province, China (GJJ13081).
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Authors’ Affiliations
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