Abstract
In this paper, the quasi-partial b-metric space is defined and general fixed point theorems on this space are discussed with examples.
Similar content being viewed by others
1 Introduction
A generalization of the metric space can be obtained as a partial-metric space by replacing the condition \(d(x,x)=0\) with the condition \(d(x,x)\le d(x,y)\) for all x, y in the definition of the metric. In the year 1993, Czerwik [1] introduced the concept of a b-metric space as another generalization of the concept of metric space. Several authors have focused on fixed point theorems for a metric space, a partial-metric space, quasi-partial metric space and a partial b-metric space. For further information on the subject see [2–16].
The concept of a quasi-partial-metric space was introduced by Karapınar et al. [17]. He studied some fixed point theorems on these spaces whereas Shatanawi and Pitea [18] studied some coupled fixed point theorems on quasi-partial-metric spaces.
The aim of this paper is to introduce the concept of quasi-partial b-metric spaces which is a generalization of the concept of quasi-partial-metric spaces. The fixed point results are proved in setting of such spaces and some examples are given to verify the effectiveness of the main results.
2 Preliminaries
We begin the section with some basic definitions and concepts.
Definition 2.1
([17])
A quasi-partial metric on a non-empty set X is a function \(q:X\times X\rightarrow \mathbb{R}^{+}\), satisfying
- (QPM1):
-
If \(q(x,x)=q(x,y)=q(y,y)\), then \(x=y\).
- (QPM2):
-
\(q(x,x)\le q(x,y)\).
- (QPM3):
-
\(q(x,x)\le q(y,x)\).
- (QPM4):
-
\(q(x,y)+q(z,z)\le q(x,z)+q(z,y)\) for all \(x,y,z\in X\).
A quasi-partial-metric space is a pair \((X, q)\) such that X is a non-empty set and q is a quasi-partial metric on X.
Let q be a quasi-partial metric on the set X. Then
Lemma 2.1
([17])
For a quasi-partial metric q on X,
Lemmas 2.2
-
(A)
A sequence \(\{x_{n} \} \) is Cauchy in a partial-metric space \((X,p)\) if and only if \(\{x_{n} \} \) is Cauchy in the (corresponding) metric space \((X,d_{p})\).
-
(B)
A partial-metric space \((X,p)\) is complete if and only if the (corresponding) metric space \((X,d_{p})\) is complete. Moreover,
$$\lim_{n\to\infty} d_{p} (x,x_{n})=0\quad \Leftrightarrow\quad p(x,x)=\lim_{n\to\infty} p(x,x_{n})=\lim _{n,m\to\infty} p(x_{n} ,x_{m}). $$
Lemma 2.3
([17])
Let \((X,q)\) be a quasi-partial metric space, let \((X,p_{q})\) be the corresponding partial-metric space, and let \((X,d_{p_{q} })\) be the corresponding metric space. Then the following statements are equivalent:
-
(A)
The sequence \(\{x_{n} \} \) is Cauchy in \((X,q)\) and \((X,q)\) is complete.
-
(B)
The sequence \(\{x_{n} \} \) is Cauchy in \((X,p_{q})\) and \((X,p_{q})\) is complete.
-
(C)
The sequence \(\{x_{n} \} \) is Cauchy in \((X,d_{p_{q} })\) and \((X,d_{p_{q} })\) is complete.
Also,
Definition 2.2
([17])
If \(T:X\to X\) is any map on X, \(O(x)=\{x,Tx,T^{2} x,\ldots\}\) is called the orbit of x. A mapping \(G:X\to{\mathbb{R}}^{+} \) is T-orbitally lower semi-continuous at x if \(\{x_{n} \} \) is a sequence in \(O(x)\) and \(\lim x_{n} =z\) implies \(G(z)\le\liminf G(x_{n})\).
3 Quasi-partial b-metric space
We introduce the concept of quasi-partial b-metric space here.
Definition 3.1
A quasi-partial b-metric on a non-empty set X is a mapping \(qp_{b} :X\times X\to{\mathbb{R}}^{+}\) such that for some real number \(s \geq1\) and all \(x, y, z\in X\):
- (QPb1):
-
\(qp_{b} (x,x)=qp_{b} (x,y)=qp_{b} (y,y)\Rightarrow x=y\),
- (QPb2):
-
\(qp_{b} (x,x)\le qp_{b} (x,y)\),
- (QPb3):
-
\(qp_{b} (x,x)\le qp_{b} (y,x)\),
- (QPb4):
-
\(qp_{b} (x,y)\le s[qp_{b} (x,z)+qp_{b} (y,z)]-qp_{b} (z,z)\).
A quasi-partial b-metric space is a pair \((X,qp_{b})\) such that X is a non-empty set and \((X,qp_{b})\) is a quasi partial b-metric on X. The number s is called the coefficient of \((X,qp_{b})\).
For a quasi-partial b-metric space \((X,qp_{b})\), the function \(d_{qp_{b} } :X\times X\to{\mathbb{R}}^{+} \) defined by
Example 3.1
Let \(X=[0,1]\).
Define \(qp_{b}(x,y)=|x-y|+x\). Here
Again, \(qp_{b}(x,x)\le qp_{b}(x,y)\) as \(x\le|x-y|+x\) and similarly, \(qp_{b}(x,x)\le qp_{b}(y,x)\) as \(x\le|y-x|+y\) for \(0< x<y\).
Also \(qp_{b}(x,y)+qp_{b}(z,z)\le s[qp_{b}(x,z)+qp_{b}(z,y)]\) as
It can be observed that
So \((X,qp_{b})\) is a quasi-partial b-metric space with \(s\ge1\).
Example 3.2
Let \(X=[1, \infty)\).
Define \(qp_{b} :X\times X\to{\mathbb{R}}^{+} \) as \(qp_{b} (x,y)=\ln (xy)\). Then \((X,qp_{b})\) is a quasi-partial b-metric space.
Let \(qp_{b} (x,x)=qp_{b} (x,y)=qp_{b} (y,y) \Rightarrow \ln(x^{2})=\ln(xy)=\ln(y^{2}) \Rightarrow x=y\).
Let \(x,y\in X\). Without loss of generality \(x \leq y \Rightarrow\ln x\le\ln y \Rightarrow2\ln x\le\ln x+\ln y \Rightarrow\ln(x^{2})\le\ln x+\ln y\).
Thus, \(qp_{b} (x,x)\le qp_{b} (x,y)\).
Similarly \(qp_{b} (x,x)\le qp_{b} (y,x)\).
For (QPb4) we have
Example 3.3
Let \(X= [0, \frac{\pi}{4} ]\) and define \(qp_{b}:X\times X\to \mathbb{R}^{+}\) as
Then \((X,qp_{b})\) is a quasi-partial b-metric space.
Lemma 3.4
Let \((X,qp_{b})\) be a quasi-partial b-metric space. Then the following hold:
-
(A)
If \(qp_{b} (x,y)=0\) then \(x = y\).
-
(B)
If \(x\ne y\), then \(qp_{b} (x,y)>0\) and \(qp_{b} (y,x)>0\).
The proof is similar to the case of quasi-partial-metric space [17].
Lemma 3.5
Every quasi-partial space is a quasi-partial b-metric space. But the converse does not need to be true.
Definition 3.2
Let \((X,qp_{b})\) be a quasi-partial b-metric. Then:
-
(i)
A sequence \(\{x_{n} \} \subset X\) converges to \(x\in X\) if and only if
$$qp_{b} (x,x)=\lim_{n\to\infty} qp_{b} (x,x_{n})=\lim_{n\to\infty} qp_{b} (x_{n} ,x). $$ -
(ii)
A sequence \(\{x_{n} \} \subset X\) is called a Cauchy sequence if and only if
$$\lim_{n,m\to\infty} qp_{b} (x_{n} ,x_{m}) \quad\mbox{and}\quad \lim_{n,m\to\infty}qp_{b} (x_{m} ,x_{n}) \mbox{ exist (and are finite)}. $$ -
(iii)
The quasi-partial b-metric space \((X,qp_{b})\) is said to be complete if every Cauchy sequence \(\{x_{n} \}\subset X\) converges with respect to \(\tau_{qp_{b} } \) to a point \(x\in X\) such that
$$qp_{b} (x,x)=\lim_{n,m\to\infty}qp_{b} (x_{m} ,x_{n})=\lim_{n,m\to\infty} qp_{b} (x_{n} ,x_{m}). $$ -
(iv)
A mapping \(f:X\to X\) is said to be continuous at \(x_{0} \in X\) if, for every \(\varepsilon>0\), there exists \(\delta>0\) such that \(f(B(x_{0} ,\delta))\subset B(f(x_{0}),\varepsilon)\).
Lemma 3.6
Let \((X,qp_{b})\) be a quasi-partial b-metric space and \((X,d_{qp_{b} })\) be the corresponding b-metric space. Then \((X,d_{qp_{b} })\) is complete if \((X,qp_{b})\) is complete.
Proof
Since \((X,qp_{b})\) is complete, every Cauchy sequence \(\{x_{n} \} \) in X converges with respect to \(\tau_{qp_{b} } \) to a point \(x\in X\) such that
Consider a Cauchy sequence \(\{x_{n} \} \) in \((X,d_{qp_{b} })\). We will show that \(\{x_{n} \} \) is Cauchy in \((X,qp_{b})\). Since \(\{x_{n} \} \) is Cauchy in \((X,d_{qp_{b} })\), \(\lim_{n,m\to\infty} d_{qp_{b} } (x_{n} ,x_{m})\) exists and is finite.
Also, \(d_{qp_{b} } (x_{n} ,x_{m})=qp_{b} (x_{n} ,x_{m})+qp_{b} (x_{m} ,x_{n})-qp_{b} (x_{n} ,x_{n})-qp_{b} (x_{m} ,x_{m})\).
Clearly, \(\lim_{n,m\to\infty} qp_{b} (x_{n} ,x_{m})\) and \(\lim_{n,m\to\infty} qp_{b} (x_{m} ,x_{n})\) exist and are finite.
Therefore, \(\{x_{n} \} \) is a Cauchy sequence in \((X,qp_{b})\). Now, since \((X,qp_{b})\) is complete, the sequence \(\{x_{n} \} \) converges with respect to \(\tau_{qp_{b} } \)to a point \(x\in X\) such that (1) holds.
For \(\{x_{n} \} \) to be convergent in \((X,d_{qp_{b} })\) we will show that \(d_{qp_{b} } (x,x)=\lim_{n\to\infty} d_{qp_{b} } (x,x_{n})\).
If follows from the definition of \(d_{qp_{b} } \) that \(d_{qp_{b} } (x,x)=0\). Also,
Hence, \(d_{qp_{b} } (x,x)=\lim_{n\to\infty} d_{qp_{b} } (x,x_{n})\). □
In [17] Karapınar et al. proved a fixed point theorem on quasi-partial-metric space. Motivated by this, we have generalized the results on a quasi-partial b-metric space.
4 The main results
Theorem 4.1
Let \((X,qp_{b})\) be a quasi-partial b-metric space, and let \(T:X\to X\). Then the following hold:
-
(A)
There exists \(\phi:X\to{\mathbb{R}}^{+} \) such that
$$\begin{aligned} &qp_{b} (x,Tx)\le\phi(x)-\phi(Tx) \quad\textit{for all } x \in X \quad\textit{if and only if} \\ &\quad\sum_{n=0}^{\infty} qp_{b} \bigl(T^{n} x,T^{n+1} x\bigr) \textit{ converges for all } x\in X. \end{aligned}$$ -
(B)
There exists \(\phi:X\to{\mathbb{R}}^{+} \) such that
$$\begin{aligned} &qp_{b} (x,Tx)\le\phi(x)-\phi(Tx) \quad\textit{for all }x\in O(x)\quad\textit{if and only if} \\ &\quad\sum_{n=0}^{\infty} qp_{b} \bigl(T^{n} x,T^{n+1} x\bigr) \textit{ converges for all }x\in O(x). \end{aligned}$$
Proof
(A) Let \(x\in X\), and let
Define the sequence \(\{x_{n} \} _{n=1}^{\infty} \) in the following way:
Set \(z_{n} (x)=\sum_{k=0}^{n} qp_{b} (x_{k} ,x_{k+1})=\sum_{k=0}^{n} qp_{b} (T^{k} x_{0} ,T^{k+1} x_{0})\). Then
Thus, (2) implies that \(\{z_{n} (x)\} \) is bounded. Also \(\{ z_{n} (x)\} \) is non-decreasing and hence convergent. Therefore, \(\sum_{n=0}^{\infty} qp_{b} (T^{n} x,T^{n+1} x)\) converges.
Conversely, define
Then
Using these definitions, we get
Since \(\sum_{n=0}^{\infty} qp_{b} (T^{n} x, T^{n+1} x)\) converges for all \(x \in X\),
Letting \(n\to\infty\) in (3) gives \(qp_{b} (x,Tx)=\phi (x)-\phi(Tx)\).
(B) It can easily be proved using part (A). □
Example 4.1
Let \(X=[0, 1]\). Define \(qp_{b} (x,y)=|x-y| + |x|\).
Then \(qp_{b} (x,y)\) satisfies all conditions of quasi-partial b-metric space. It is also quasi-partial metric. But for \(x\ne y\), \(qp_{b} (x,y)\ne qp_{b} (y,x)\) and \(qp_{b} (x,x)\ne0\) for \(x\ne0\). So \(qp_{b} \) is not a partial metric or a quasi-metric. Define \(T:X\to X\) as \(Tx=\frac{x}{3} \) for all \(x\in X\). Then the series \(\sum_{n=0}^{\infty} qp_{b} (T^{n} x,T^{n+1} x)\) is convergent. Indeed,
Then the conditions of Theorem 4.1 are satisfied for \(\phi(x)=\frac {5x}{2} \). Indeed
The next result gives conditions for the existence of fixed points of operators on quasi-partial b-metric space.
Theorem 4.2
Let \((X, qp_{b})\) and \((Y, qp_{b})\) be complete quasi-partial b-metric spaces. Let also \(T:X\to X\), \(R: X\to Y\), and \(\phi:R(X)\to{\mathbb{R}}^{+} \). If there exist \(x\in X\) and \(c>0\) such that
for all \(y\in O(x)\), then the following hold:
-
(A)
\(\lim_{n\to\infty} T^{n} x=z\) exists.
-
(B)
\(Tz=z\) if and only if \(G(x)=qp_{b} (x, Tx)\) is T-orbitally lower semi-continuous at x.
-
(C)
\(qp_{b} (x, T^{n} x)\le s^{n-1} \phi(Rx)\).
-
(D)
For \(m > n\), \(qp_{b} (T^{n} x, T^{m} x)\le s^{m-n} [\phi(RT^{n} x)]\).
Proof
(A) Let \(x\in X\). Define the sequence \(\{x_{n} \} _{n=1}^{\infty} \) as follows:
We will show that \(\{x_{n} \} _{n=1}^{\infty} \) is Cauchy.
Using (QPb4), we get
and, similarly,
Now,
On generalization, we get
Set \(z_{n} (x)=\sum_{k=0}^{n} s^{m-k} \{qp_{b} (T^{k} x, T^{k+1} x)\} \).
From (4) we have
Thus, \(\sum_{k=0}^{\infty} s^{m-k} \{ qp_{b} (T^{k} x, T^{k+1} x)\}\) is convergent.
Taking the limit as \(n,m\to\infty\) in (7), we get
Using similar arguments,
Thus the sequence \(\{x_{n} \} \) is Cauchy in \((X,qp_{b})\). Since \((X,qp_{b})\)is complete, \((X,d_{qp_{b} })\) is also complete by Lemma 2.3, and hence \(\lim_{n\to\infty} d_{qp_{b} } (T^{n} x,z)=0\), \(\lim_{n\to\infty} T^{n} x=z\).
Further, \(\lim_{n\to\infty} qp_{b} (T^{n} x, T^{n+1} x)=0\) and hence \(\lim_{n\to\infty} qp_{b} (T^{n} x, T^{n+1} x)=qp_{b} (z,z)=0\).
(B) Assume that \(Tz=z\) and that \(x_{n} \) is a sequence in \(O(x)\) with \(x_{n} \to z\).
By Lemma 3.6,
Then \(G(z)=qp_{b} (z,Tz)=qp_{b} (z,z)\le\lim_{n\to\infty} \inf qp_{b} (x_{n} ,Tx_{n})=\lim_{n\to\infty} \inf G(x_{n})\).
Thus G is T-orbitally lower semi-continuous at x.
Conversely, suppose that \(x_{n} =T^{n} x\to z\) and that G is T-orbitally lower semi-continuous at x. Then
By Lemma 3.4, we have \(Tz=z\).
(C) We have, from (QPb4) and (4),
On generalization, we get
(D) From (7) we get
Note that
Here, \(0\le qp_{b} (x_{n},x_{m}) =qp_{b} (T^{n} x, T^{m} x)\le s^{m-n} \phi(RT^{n} x)\) for \(m>n\). □
Example 4.2
Let \(X=Y=[0,1]\). Define \(qp_{b}(x,y)=|x-y|+x\). Then \(qp_{b}\) is a quasi-partial b-metric with \(s=1\). Also define \(T:X\to X\) as \(T(x)=\frac{x}{3}\); \(R:X\to Y\) as \(R(x)=3x\), and \(\phi:R(X)\to \mathbb{R}^{+}\) as \(\phi(x)=3x\). Then for \(c=1\) and \(x\in [0,1]\) we have
We now prove that (A), (B), (C), and (D) of the above theorem hold:
(A) \(\lim_{n\to\infty}T^{n}x=\lim_{n\to\infty}\frac {x}{3^{n}}=0=z\) (say).
So \(\lim_{n\to\infty}T^{n}x=z\) exists.
(B) By (A) part above, \(z=0\).
Therefore \(T(z)=T(0)=0=z\) holds trivially.
Hence whenever \(G(x)=qp_{b}(x,Tx)\) is T-orbitally lower semi-continuous at x then \(Tz=z\).
Conversely, let \(Tz=z\) and we show that G is T-orbitally lower semi-continuous at x, i.e.,
Let \(\{x_{n}\}\subseteq O(x)\) be a sequence converging to z. Then
Hence \(G(z)=\lim\inf G(x_{n})\).
\(\begin{array}{ll} (\mathrm{C})\ \displaystyle qp_{b}\bigl(x,T^{n}x\bigr) &\displaystyle=qp_{b} \biggl(x, \frac{x}{3^{n}} \biggr)=\biggl \vert x-\frac{x}{3^{n}}\biggr \vert +x=x \biggl(2-\frac{1}{3^{n}} \biggr)< x(9) \quad \forall n\in N\\ \displaystyle&\displaystyle=\phi(3x) =s^{n-1}\phi(Rx) \quad\mbox{where } s=1. \end{array}\)
(D) Let \(m>n\) then
Corollary 4.3
Let \((X, qp_{b})\) be a complete quasi-partial b-metric space. Let \(T:X\to X\) and \(\phi:X\to{\mathbb{R}}^{+} \). Suppose that there exists \(x\in X\) such that
Then the following hold:
-
(A)
\(\lim_{n\to\infty} T^{n} x=z\) exists.
-
(B)
\(Tz=z\) if and only if \(G(x)=qp_{b} (x, Tx)\) is T-orbitally lower semi-continuous at x.
-
(C)
\(qp_{b} (x, T^{n} x)\le s^{n-1} \phi(x)\).
-
(D)
For \(m > n\), \(qp_{b} (T^{n} x, T^{m} x)\le s^{m-n} \phi(T^{n} x)\).
Proof
Take \(Y = X\), \(R = I\), and \(c = 1\) in Theorem 4.2. □
Corollary 4.4
Let \((X, qp_{b})\) be a complete quasi-partial b-metric space, and let \(0< k<1\). Suppose that \(T:X\to X\) and that there exists \(x\in X\) such that
Then the following hold:
-
(A)
\(\lim_{n\to\infty} T^{n} x=z\) exists.
-
(B)
\(Tz=z\) if and only if \(G(x)=qp_{b} (x, Tx)\) is T-orbitally lower semi-continuous at x.
-
(C)
\(qp_{b} (x, T^{n} x)\le\frac{s^{n-1} }{1-k} qp_{b} (x, Tx)\).
Proof
Set \(\phi(y)=\frac{1}{1-k} qp_{b} (y, Ty)\) for \(y\in O(x)\).
Let \(y=T^{n} x\) in (16). Then
and
Thus, \(qp_{b} (T^{n} x, T^{n+1} x)\le\frac{1}{1-k} [qp_{b} (T^{n} x, T^{n+1} x)-qp_{b} (T^{n+1} x, T^{n+2} x)]\) or \(qp_{b} (y,Ty)\le[\phi(y)-\phi(Ty)]\).
(A)-(C) follow immediately from Corollary 4.3. □
Corollary 4.5
Let \((X,qp_{b})\) be a complete quasi-partial b-metric space where \(qp_{b} \) is continuous. Let \(T:X\to X\) and \(\phi:X\to{\mathbb{R}}^{+} \) is continuous. Suppose that there exists \(x\in X\) such that
Then the following hold:
-
(A)
\(\lim_{n\to\infty} T^{n} x=z\) exists.
-
(B)
\(q_{p} (z,z)\le s\phi(z)\).
Proof
In Theorem 4.2(D) taking \(m = n + 1\), \(R = I\), \(c = 1\), and \(Y = X\),
Now taking \(\lim n\to\infty\)
□
References
Czerwik, S: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 1, 5-11 (1993)
Mukheimer, A: α-ψ-ϕ-Contractive mappings in ordered partial b-metric spaces. J. Nonlinear Sci. Appl. 7, 168-179 (2014)
Shatanawi, W: On ω-compatible mappings and common coupled coincidence point in cone metric spaces. Appl. Math. Lett. 25, 925-931 (2012)
Bhaskar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379-1393 (2006)
Lakshmikantham, V, Ćirić, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70, 4341-4349 (2009)
Hicks, TL: Fixed point theorems for quasi-metric spaces. Math. Jpn. 33(2), 231-236 (1988)
Karapınar, E: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011, Article ID 4 (2011)
Ali, MU: Mizoguchi-Takahashi’s type common fixed point theorem. J. Egypt. Math. Soc. 22, 272-274 (2014)
Bakhtin, IA: The contraction principle in quasimetric spaces. In: Functional Analysis, vol. 30, pp. 26-37 (1989)
Bota, M-F, Karapınar, E, Mleşniţe, O: Ulam-Hyers stability results for fixed point problems via α-ψ-contractive mapping in (b)-metric space. Abstr. Appl. Anal. 2013, Article ID 825293 (2013)
Bota, M-F, Karapınar, E: A note on ‘Some results on multi-valued weakly Jungck mappings in b-metric space’. Cent. Eur. J. Math. 11(9), 1711-1712 (2013)
Aydi, H, Bota, M-F, Karapınar, E, Moradi, S: A common fixed point for weak ϕ-contractions ON b-metric spaces. Fixed Point Theory 13(2), 337-346 (2012)
Aydi, H, Bota, M-F, Karapınar, E, Mitrović, S: A fixed point theorem for set-valued quasi-contractions in b-metric spaces. Fixed Point Theory Appl. 2012, Article ID 88 (2012)
Latif, A, Al-Mezel, SA: Fixed point results in quasi metrics spaces. Fixed Point Theory Appl. 2011, Article ID 178306 (2011)
Shukla, S: Partial b-metric spaces and fixed point theorems. Mediterr. J. Math. 11, 703-711 (2014)
Caristi, J: Fixed point theorems for mapping satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241-251 (1976)
Karapınar, E, Erhan, ÍM, Özturk, A: Fixed point theorems on quasi-partial metric spaces. Math. Comput. Model. 57, 2442-2448 (2013)
Shatanawi, W, Pitea, A: Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl. 2013, Article ID 153 (2013). doi:10.1186/1637-1812-2013-153
Matthews, SG: Partial metric topology, general topology and its applications. Ann. N.Y. Acad. Sci. 728, 183-197 (1994)
Altun, I, Erduran, A: Fixed point theorems, for monotone mappings on partial metric spaces. Fixed Point Theory Appl. 2011, Article ID 508730 (2011)
Matthews, SG: Partial Metric Topology. Research Report 212, Department of Computer Science, University of Warwick (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
About this article
Cite this article
Gupta, A., Gautam, P. Quasi-partial b-metric spaces and some related fixed point theorems. Fixed Point Theory Appl 2015, 18 (2015). https://doi.org/10.1186/s13663-015-0260-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13663-015-0260-2