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Common fixed point theorems for generalized cyclic contraction pairs in bmetric spaces with applications
Fixed Point Theory and Applications volume 2015, Article number: 164 (2015)
Abstract
In this paper, we introduce the notion of generalized cyclic contraction pairs in bmetric spaces and establish some fixed point theorems for such pairs. Also, we give some examples to illustrate the main results which properly generalizes some results given by some authors in literature. Further, by using the main results, we prove some common fixed point results for generalized contraction pairs in partially ordered bmetric spaces. Our results generalize and improve the result of Shatanawi and Postolache (Fixed Point Theory Appl. 2013:60, 2013) and several wellknown results given by some authors in metric and bmetric spaces.
Introduction
Fixed point theory plays a basic role in applications of many branches of mathematics. Finding fixed points of generalized contraction mappings has become the focus of strong research activity in fixed point theory. Recently, many authors have published many papers on fixed point theory and applications in several ways. One of the recently popular topics in fixed point theory is to show the existence of fixed points of cyclic contraction mappings in several spaces. In 2003, Kirk et al. [1] introduced concepts of cyclic mappings and cyclic contraction mappings and also gave some interesting fixed point theorems for these mappings. Later, several mathematicians have been studying fixed point results for cyclic mappings satisfying generalized contraction conditions (see in [2–5]).
In 2013, Shatanawi and Postolache [6] introduced the notion of a generalized cyclic contraction for the pair of selfmappings in partially ordered metric spaces and proved some common fixed point theorems for such a pair by using the idea of altering distance functions due to Khan et al. [7].
On the other hand, in 1989, Bakhtin [8] introduced the concept of bmetric spaces as a generalization of metric spaces and also proved Banach’s contraction principle in bmetric spaces, which is a generalization of Banach’s contraction principle in metric spaces. Afterward, many mathematicians have studied fixed point results for singlevalued and multivalued mappings in bmetric spaces (see [9–13]).
In this paper, we introduce the concept of new generalized cyclic contraction pairs in bmetric spaces and establish some fixed point theorems for such pairs in the setting of bmetric spaces. Also, we give some examples to illustrate that our results properly generalize some results given by some authors in literature. Further, by using our main results, we give some common fixed results in partially ordered bmetric spaces. Our result generalizes and improves the corresponding results of Shatanawi and Postolache [6] and several wellknown results of fixed point and common fixed point theorems given by some authors in metric and bmetric spaces.
Preliminaries
Throughout this paper, we denote by \(\mathbb{N}\), \(\mathbb{R}_{+}\) and \(\mathbb{R}\) the sets of positive integers, nonnegative real numbers and real numbers, respectively.
In 1984, Khan et al. [7] introduced the concept of an altering distance function as follows.
Definition 2.1
The function \(\varphi: [0,\infty) \rightarrow[0,\infty)\) is called an altering distance function if the following properties hold:

(1)
φ is continuous and nondecreasing;

(2)
\(\varphi(t) = 0\) if and only if \(t = 0\).
The following examples illustrate the definition.
Example 2.2
Let \(\varphi_{1}, \varphi_{2} : [0,\infty) \rightarrow[0,\infty)\) be defined by
for all \(t \in[0,\infty)\), where \(k \in[0,1)\) and \(l \in(0,\infty)\). Then the functions \(\varphi_{1}\) and \(\varphi_{2}\) are altering distance functions (see the geometry of the functions \(\varphi_{1}\) and \(\varphi_{2}\) in Figure 1).
In 2003, Kirk et al. [1] introduced the concepts of cyclic mappings and cyclic contractions as follows.
Definition 2.3
([1])
Let A and B be nonempty subsets of a metric space \((X,d)\). A mapping \(f:A\cup B \to A\cup B\) is said to be cyclic if \(f(A)\subseteq B\) and \(f(B)\subseteq A\).
Definition 2.4
([1])
Let A and B be nonempty subsets of a metric space \((X,d)\). A mapping \(f:A\cup B \to A\cup B\) is called a cyclic contraction if there exists \(k\in[0,1)\) such that
for all \(x\in A\) and \(y\in B\).
On the other hand, Bakhtin [8] introduced the concept of a bmetric space as follows.
Definition 2.5
([8])
Let X be a nonempty set and \(s\geq1\). Suppose that the mapping \(d : X \times X \rightarrow\mathbb{R}_{+}\) satisfies the following conditions:

(1)
\(0 \leq d(x,y)\) for all \(x,y \in X\) and \(d(x,y) = 0\) if and only if \(x = y\);

(2)
\(d(x, y)= d(y,x)\) for all \(x,y \in X\);

(3)
\(d(x,y) \leq s[d(x,z)+ d(z,y)]\) for all \(x,y,z, \in X\).
Then \((X,d)\) is called a bmetric space with coefficient s.
Every metric space is a bmetric space with \(s=1\) and so the class of bmetric spaces is larger than the class of metric spaces. In general, a bmetric space does not necessarily need to be a metric space.
Now, we give some known examples of a bmetric which show that a bmetric space is a real generalization of metric spaces as follows.
Example 2.6
Let \(X=\mathbb{R}\) and a mapping \(d : X \times X \rightarrow\mathbb {R}_{+}\) be defined by
for all \(x,y \in X\). Then \((X,d)\) is a bmetric space with coefficient \(s=2\).
Example 2.7
The set \(l_{p}(\mathbb{R})\) with \(0 < p < 1\), where
together with the mapping \(d : l_{p}(\mathbb{R})\times l_{p}(\mathbb {R})\rightarrow\mathbb{R}_{+}\) defined by
for each \(x = \{x_{n}\}, y = \{y_{n}\} \in l_{p}(\mathbb{R})\) is a bmetric space with coefficient \(s = 2^{\frac{1}{p}} > 1\). The above result also holds for the general case \(l_{p}(X)\) with \(0 < p < 1\), where X is a Banach space.
Example 2.8
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 \(\int_{0}^{1} x(t)^{p} \,dt < 1\) together with the mapping \(d:L_{p}[0, 1]\times L_{p}[0, 1] \rightarrow \mathbb{R}_{+}\) defined by
for each \(x, y \in L_{p}[0, 1]\) is a bmetric space with constant \(s = 2^{\frac{1}{p}}\).
Next, we give the concepts of convergence, a Cauchy sequence, bcontinuity, and completeness and closedness in a bmetric space.
Definition 2.9
([10])
Let \((X, d)\) be a bmetric space. Then a sequence \(\{x_{n}\}\) in X is called:

(1)
bconvergent if there exists \(x\in X\) such that \(d(x_{n},x)\rightarrow0\) as \(n\rightarrow\infty\); in this case, we write \(\lim_{n\rightarrow\infty} x_{n} = x\);

(2)
a bCauchy sequence if \(d(x_{n}, x_{m}) \rightarrow0\) as \(n,m \rightarrow\infty\).
Proposition 2.10
([10])
In a bmetric space \((X, d)\), the following assertions hold:

(1)
a bconvergent sequence has a unique limit;

(2)
each bconvergent sequence is a bCauchy sequence;

(3)
in general, a bmetric is not continuous.
We need the following lemma as regards bconvergent sequences in the proof of our results.
Lemma 2.11
([14])
Let \((X,d)\) be a bmetric space with coefficient \(s\geq1\) and let \(\{x_{n}\}\), \(\{y_{n}\}\) be bconvergent to the points \(x, y \in X\), respectively. Then we have
In particular, if \(x=y\), then we have \(\lim_{n\rightarrow\infty} d(x_{n},y_{n}) = 0\). Moreover, for each \(z\in X\),
Definition 2.12
([10])
Let \((X, d)\) and \((X', d')\) be two bmetric spaces.

(1)
The space \((X, d)\) is bcomplete if every bCauchy sequence in X bconverges;

(2)
a function \(f : X\rightarrow X'\) is bcontinuous at a point \(x\in X\) if it is bsequentially continuous at x, that is, whenever \(\{x_{n}\}\) is bconvergent to x, \(\{fx_{n}\}\) is bconvergent to fx.
Definition 2.13
([10])
Let Y be a nonempty subset of a bmetric space \((X, d)\). The closure Y̅ of Y is the set of limits of all bconvergent sequences of points in Y, i.e.,
Definition 2.14
([10])
Let \((X, d)\) be a bmetric space. Then a subset \(Y \subseteq X\) is said to be closed if and only if, for each sequence \(\{x_{n}\}\) in Y which bconverges to a point x, we have \(x \in Y\) (i.e., \(\overline{Y}= Y\)).
In 2014, Sintunavarat [15] (see also [16]) introduced the useful concept of transitivity for mappings as follows.
Definition 2.15
Let X be a nonempty set. The mapping \(\alpha: X \times X \rightarrow[0,\infty)\) is said to be transitive if, for all \(x, y, z\in X\),
Main results
Let \((X,d)\) be a bmetric space with coefficient \(s\geq1\) and \(f, g : X \rightarrow X\) be two selfmappings. Throughout this paper, unless otherwise stated, for all \(x,y \in X\), let
If \(s=1\), we write \(M(x,y)\) instead \(M_{s}(x,y)\), that is,
Definition 3.1
Let A, B be two nonempty closed subsets of a bmetric space \((X,d)\) with coefficient \(s \geq1\), \(\alpha:X \times X \rightarrow [0,\infty)\), \(\psi: [0,\infty) \rightarrow[0,\infty)\) and \(f, g : X \to X\) be four mappings. The pair \((f,g)\) is called an cyclic α \((\psi ,A,B)_{s}\) contraction if

(1)
ψ is an altering distance function;

(2)
\(A\cup B\) has a cyclic representation w.r.t. the pair \((f,g)\), that is, \(f(A) \subseteq B\), \(g(B) \subseteq A\), and \(X=A\cup B\);

(3)
there exists \(0<\delta<1\) such that the following condition holds:
$$ \begin{aligned} &x \in A, y \in B \quad\mbox{with } \alpha(x,y)\geq1 \mbox{ or } \alpha(y,x)\geq1 \\ &\quad\Longrightarrow\quad \psi \bigl(s^{3}d(fx,gy) \bigr) \leq\delta\psi \bigl(M_{s}(x,y) \bigr). \end{aligned} $$(3.1)
Definition 3.2
Let A, B be two nonempty closed subsets of a bmetric space \((X,d)\) with \(X=A\cup B\) and \(\alpha:X \times X \rightarrow[0,\infty)\) and \(f, g : X \to X\) be three mappings. The pair \((f,g)\) is said to be α \((A,B)\) weakly increasing if \(\alpha(fx,gfx)\geq1\) for all \(x \in A\) and \(\alpha(gx,fgx)\geq1\) for all \(x \in B\).
Now, we give the main results in this section.
Theorem 3.3
Let \((X,d)\) be a complete bmetric space with coefficient \(s \geq1\) and A, B be nonempty closed subsets of X. Suppose that \(\alpha:X \times X \rightarrow[0,\infty)\), \(\psi: [0,\infty) \rightarrow[0,\infty)\) and \(f, g : X \to X\) are four mappings such that the pair \((f,g)\) is α\((A,B)\)weakly increasing and the following conditions hold:

(1)
the pair \((f,g)\) is a cyclic α\((\psi,A,B)_{s}\)contraction;

(2)
f or g is bcontinuous;

(3)
α is a transitive mapping;

(4)
if \(\{x_{n}\}\) is sequence in X such that \(\alpha (x_{n},x_{n+1})\geq1\) and \(x_{n} \to z\) as \(n \to\infty\), then \(\alpha(z,z)\geq1\).
Then f and g have a common fixed point in \(A \cap B\).
Proof
Choose \(x_{0} \in A\). Let \(x_{1} := fx_{0}\). Since \(f(A) \subseteq B\), we have \(x_{1} \in B\). Also, let \(x_{2} := gx_{1}\). Since \(g(B) \subseteq A\), we have \(x_{2} \in A\). Continuing this process, we can construct a sequence \(\{x_{n}\}\) in X such that
for all \(n\in\mathbb{N}\cup\{0\}\). Since f and g are α\((A,B)\)weakly increasing, we have \(\alpha(fx_{0},gfx_{0})\geq1\) and \(\alpha(gx_{1},fgx_{1})\geq1\). This implies that \(\alpha(x_{1},x_{2})\geq1\) and \(\alpha(x_{2},x_{3})\geq1\). Repeating this process, we obtain
for all \(n \in\mathbb{N} \cup\{0\}\). From (3.1), we have
for all \(n \in\mathbb{N} \cup\{0\}\).
Now, we complete the proof by the following three steps:
Step I. We prove that \(\lim_{k\rightarrow\infty} d(x_{k},x_{k+1}) = 0\). For each \(k \in\mathbb{N} \cup\{0\}\), we define \(d_{k} := d(x_{k},x_{k+1})\). Now, we assume that \(d_{k_{0}} = 0\) for some \(k_{0}\in\mathbb{N} \cup\{ 0\}\). This implies that \(x_{k_{0}} = x_{{k_{0}}+1}\). If \(k_{0} = 2n\) for some \(n \in\mathbb{N}\), then \(x_{2n} = x_{2n+1}\).
Next, we show that \(x_{2n+1} = x_{2n+2}\). Since \(\alpha (x_{2n},x_{2n+1}) \geq1\), we have
where
Therefore, from (3.3), it follows that
Since \(\delta<1\), we have \(\psi(s^{3}d(x_{2n+1},x_{2n+2})) = 0\) and hence \(x_{2n+1} = x_{2n+2}\).
Similarly, if \(k_{0} = 2n+1\) for some \(n\in\mathbb{N}\cup\{0\}\), then \(x_{2n+1}=x_{2n+2}\) gives \(x_{2n+2}=x_{2n+3}\). Consequently, the sequence \(\{d_{k}\}\) becomes constant for \(k\geq k_{0}\) and hence \(\lim_{k\rightarrow\infty} d(x_{k},x_{k+1}) = 0\). This completes this step. Therefore, we suppose that
for all \(k\in\mathbb{N}\cup\{0\}\).
Next, we will show that
for all \(k \in\mathbb{N} \cup\{0\}\). Assume to the contrary that
for some \(k\in\mathbb{N} \cup\{0\}\). If k is even, then \(k=2n\) for some \(n\in\mathbb{N}\cup\{0\}\). Therefore, we have
Since \(x_{2n} \in A\), \(x_{2n+1} \in B\) and \(\alpha(x_{2n},x_{2n+1}) \geq1\), we have
where
Then we have
which is a contradiction. Thus we have
for all \(n\in\mathbb{N}\cup\{0\}\). If k is odd, then \(k=2n+1\) for some \(n\in\mathbb{N}\cup\{0\}\). Therefore, we have
Since \(x_{2n+2} \in A\), \(x_{2n+1} \in B\) and \(\alpha(x_{2n+1},x_{2n+2}) \geq1\), we have
where
Then we have
which is a contradiction. Therefore, we have
for all \(n\in\mathbb{N}\cup\{0\}\). Hence the inequality (3.5) holds and then \(\{d(x_{k},x_{k+1}) : k \in\mathbb{N}\cup\{0\}\}\) is bounded below and nonincreasing. Thus there exists \(r\geq0\) such that
and we obtain
Letting \(n \to\infty\) in (3.2), using (3.8), (3.9), and the properties of ψ, we have
Since \(\delta< 1\), we have \(\psi(s^{3}r)=0\) and hence \(r=0\). Thus we have
Step II. We will show that \(\{x_{n}\}\) is a bCauchy sequence in X. That is, for any \(\epsilon>0\), there exists \(k\in\mathbb{N}\) such that for all \(m,n\geq k\), we get \(d(x_{m},x_{n}) <\epsilon\). Assume to the contrary that there exists \(\epsilon> 0\) for which we can find two subsequences \(\{x_{m(k)}\}\) and \(\{x_{n(k)}\} \) of \(\{x_{n}\}\) such that \(n(k) > m(k) \geq k\) and:

(a)
\(m(k) \) is even and \(n(k)\) is odd;

(b)
$$ d(x_{m(k)},x_{n(k)}) \geq\epsilon; $$(3.11)

(c)
\(n(k)\) is the smallest number such that the condition (b) holds, i.e.,
$$ d(x_{m(k)},x_{n(k)1}) < \epsilon. $$(3.12)
From (3.11), (3.12), and the triangle inequality, we obtain
Letting \(k\rightarrow\infty\) in (3.13) and using (3.10), we have
From the triangle inequality, we have
and
Letting \(k\rightarrow\infty\) in (3.15) and (3.16), it follows from (3.10) and (3.14) that
and
which implies that
Again, using the above process, we have
Finally, we obtain
Taking the limit supremum as \(k\rightarrow\infty\) in (3.19), it follows from (3.10) and (3.17) that
Similarly, we have
Thus it follows from (3.20) and (3.21) that
Since α is transitive, we have
From (3.1), we have
where
Letting the limit supremum as \(k\rightarrow\infty\) in the above equation and using (3.11), (3.15), (3.17), (3.19), and (3.22), we have
Letting \(k\rightarrow\infty\) in (3.23), we have
Since \(\delta<1\), we have \(\psi(s\epsilon) = 0\) and hence \(\epsilon= 0\), which is a contradiction. Therefore, \(\{x_{n}\}\) is a bCauchy sequence in X.
Step III. We show that existence of a common fixed point of f and g. Since \((X,d)\) is a complete bmetric space and \(\{x_{n}\}\) is a bCauchy sequence in X, there exists \(z \in X\) such that
and so
Since \(\{x_{2n}\}\) is a sequence in A, A is closed, and \(x_{2n} \to z\), we have \(z \in A\). Also, since \(\{x_{2n+1}\}\) is a sequence in B, B is closed, and \(x_{2n+1} \to z\), we have \(z \in B\).
Now, we show that z is a fixed point of f and g. Without loss of generality, we may assume that f is continuous. Since \(\{x_{2n}\} \to z\), we get \(x_{2n+1}=fx_{2n} \to fz\). By the uniqueness of the limit, we have \(z=fz\).
Now, we show that \(z=gz\). Since \(z \in A\), \(z \in B\), and \(\alpha(z,z)\geq1\), we have
Since \(\delta<1\), it follows that \(d(z,gz)=0\) and hence \(z=gz\). Therefore, z is a common fixed point of f and g. This completes the proof. □
Theorem 3.3 can be proved without assuming the bcontinuity of f or the bcontinuity of g. For this instance, we assume that X satisfies the following property.
Definition 3.4
Let \((X,d)\) be a bmetric space and \(\alpha:X \times X \rightarrow [0,\infty)\) be a mapping. A space X satisfies the property (Q) if \(\{x_{n}\}\) is a sequence in X such that \(\alpha(x_{n},x_{n+1}) \geq 1\) for all \(n \in\mathbb{N}\) and \(x_{n} \to x\) as \(n \to\infty\), then \(\alpha(x_{n},x) \geq1\) for all \(n \in\mathbb{N}\).
Now, we state and prove the following result.
Theorem 3.5
Let \((X,d)\) be a complete bmetric space with coefficient \(s \geq1\) and A, B be nonempty closed subsets of X. Suppose that \(\alpha:X \times X \rightarrow[0,\infty)\), \(\psi: [0,\infty) \rightarrow[0,\infty)\) and \(f, g : X \to X\) are four mappings such that the pair \((f,g)\) is α\((A,B)\)weakly increasing and the following conditions hold:

(1)
the pair \((f,g)\) is a cyclic α\((\psi,A,B)_{s}\)contraction;

(2)
X satisfies the property (Q);

(3)
α is a transitive mapping;

(4)
if \(\{x_{n}\}\) is a sequence in X such that \(\alpha (x_{n},x_{n+1})\geq1\) for all \(n \in\mathbb{N}\) and \(x_{n} \to z\) as \(n \to\infty\), then \(\alpha(z,z)\geq1\).
Then f and g have a common fixed point in \(A \cap B\).
Proof
Now, we prove the reasoning of Theorem 3.3 step by step to construct a sequence \(\{x_{n}\}\) in X with
for all \(n \in\mathbb{N}\) and \(x_{n} \to u\) for some \(u \in X\). Since \(x_{2n} \to u\), \(x_{2n+1} \to u\), A and B are closed subsets of X, we have \(u \in A \cap B\). Using the property (Q), we have \(\alpha(x_{n},u) \geq1\) for all \(n \in\mathbb{N}\). Since \(x_{2n} \in A\), \(u \in B\), and \(\alpha(x_{2n},u) \geq1\) for all \(n \in\mathbb{N}\), we have
for all \(n \in\mathbb{N}\). Taking the limit supremum as \(n\to\infty\) in the above inequality, we have
Since \(\delta<1\) and \(s\geq1\), we have \(d(u,gu)=0\) and hence \(gu=u\). Similarly, we may show that \(fu=u\). Thus u is a common fixed point of f and g. This completes the proof. □
Taking \(f=g\) in Theorems 3.3 and 3.5, we have the following result.
Corollary 3.6
Let \((X,d)\) be a complete bmetric space with coefficient \(s \geq1\) and A, B be nonempty closed subsets of X. Suppose that \(\alpha:X \times X \rightarrow[0,\infty)\), \(\psi: [0,\infty) \rightarrow[0,\infty)\) and \(f : X \to X\) are three mappings such that \(\alpha(fx,ffx)\geq1\) for all \(x \in X\) and the following conditions hold:

(1)
ψ is altering distance, f is a cyclic mapping, and there exists \(0<\delta<1\) such that
$$\begin{aligned} &x \in A, y \in B \quad \textit{with } \alpha(x,y)\geq1 \textit{ or } \alpha(y,x) \geq1 \\ &\quad\Longrightarrow\quad \psi \bigl(s^{3}d(fx,fy) \bigr)\leq\delta \psi \biggl(\max \biggl\{ d(x,y),d(x,fx),d(y,fy),\frac{d(x,fy)+d(y,fx)}{2s} \biggr\} \biggr); \end{aligned}$$ 
(2)
f is bcontinuous;

(3)
α is a transitive mapping;

(4)
if \(\{x_{n}\}\) is a sequence in X such that \(\alpha (x_{n},x_{n+1})\geq1\) for all \(n \in\mathbb{N}\) and \(x_{n} \to z\) as \(n \to\infty\), then \(\alpha(z,z)\geq1\).
Then f has a fixed point in \(A\cap B\).
Corollary 3.7
Let \((X,d)\) be a complete bmetric space with coefficient \(s \geq1\) and A, B be nonempty closed subsets of X. Suppose that \(\alpha:X \times X \rightarrow[0,\infty)\), \(\psi: [0,\infty) \rightarrow[0,\infty)\), and \(f : X \to X\) are three mappings such that \(\alpha(fx,ffx)\geq1\) for all \(x \in X\) and the following conditions hold:

(1)
ψ is altering distance, f is a cyclic mapping, and there exists \(0<\delta<1\) such that
$$\begin{aligned} &x \in A, y \in B \quad \textit{with } \alpha(x,y)\geq1 \textit{ or } \alpha(y,x) \geq1 \\ &\quad\Longrightarrow\quad \psi \bigl(s^{3}d(fx,fy) \bigr)\leq\delta \psi \biggl(\max \biggl\{ d(x,y),d(x,fx),d(y,fy),\frac{d(x,fy)+d(y,fx)}{2s} \biggr\} \biggr); \end{aligned}$$ 
(2)
X satisfies the property (Q);

(3)
α is a transitive mapping;

(4)
if \(\{x_{n}\}\) is a sequence in X such that \(\alpha (x_{n},x_{n+1})\geq1\) for all \(n \in\mathbb{N}\) and \(x_{n} \to z\) as \(n \to\infty\) then \(\alpha(z,z)\geq1\).
Then f has a fixed point in \(A\cap B\).
Now, we give an example to illustrate the utility of Theorem 3.5.
Example 3.8
Let \(X=\mathbb{N} \cup\{\infty\}\) and \(d : X \times X \rightarrow [0,\infty)\) be defined by
Thus \((X,d)\) is a complete bmetric space with coefficient \(s=\frac{5}{2}\). Let
and
Note that A and B are nonempty closed subset of X and \(X=A \cup B\). Define three mappings \(f, g : X \rightarrow X\) and \(\alpha: X \times X \rightarrow[0,\infty)\) by
and
Also, define \(\psi: [0,\infty) \to[0,\infty)\) by \(\psi(t)=\sqrt{t}\) for all \(t \in[0,\infty)\).
Now, we show that all the conditions in Theorem 3.5 hold in this situation.
To prove that \((f,g)\) is α\((A,B)\)weakly increasing. Let \(n \in A\). Then it follows that \(fn\in B\) and so \(gfn = \infty\). Thus \(\alpha(fn,gfn) \geq1\) for all \(n \in A\). Let \(n \in B\). Then \(gn = \infty\) and \(fgn = \infty\). Thus \(\alpha(gn,fgn) \geq1\) for all \(n \in B\). Therefore, \((f,g)\) is α\((A,B)\)weakly increasing.
Next, we show that \((f,g)\) is a cyclic α\((\psi,A,B)_{s}\)contraction.

(a)
It is easy to see that ψ is an altering distance function;

(b)
since \(f(A) \subseteq\{16n : n \in\mathbb{N}\}\cup\{\infty \} = B\) and \(g(B) = \{\infty\} \subseteq A\), we conclude that \(A\cup B\) has a cyclic representation with respect to the pair \((f,g)\);

(c)
here, we show that f and g satisfy the condition (3.1). Let \(m \in A\) and \(n \in B\). We show this proof in two cases.
Case I: Assume that \(m \in A/\{\infty\}\). Then we have
Case II: Assume that \(m = \infty\). Now, we have \(d(fm,gn)=0\). Then we have nothing to prove.
From (a), (b), and (c), it follows that \((f,g)\) is a cyclic α\((\psi,A,B)_{s}\)contraction with \(\delta= \frac{5\sqrt{5}}{8\sqrt{2}} < 1\). It is easily to show that X satisfies the property (Q) and α is transitive. Moreover, the condition (4) of Theorem 3.5 holds. Thus f and g satisfy all the conditions of Theorem 3.5. Hence f and g have a common fixed point, i.e., a point ∞ is a common fixed point of f and g.
Some particular cases
In this section, we give some fixed point results on partially ordered bmetric spaces which can be regarded as consequences of the results presented in the previous section.
Now, we need the following notions and definitions for the main results in this section.
Definition 4.1
Let X be a nonempty set. Then \((X,d,\preceq)\) is called a partially ordered bmetric space if \((X,d)\) is a bmetric space and \((X,\preceq)\) is a partially ordered set.
Definition 4.2
Let A, B be two nonempty closed subsets of a complete partially ordered bmetric space \((X,d,\preceq)\) with coefficient \(s \geq1\) and \(\psi:[0,\infty) \rightarrow[0,\infty )\), \(f, g : X \to X\) be three mappings. The pair \((f,g)\) is called a cyclic \((\psi,A,B)_{s}\) contraction if

(1)
ψ is an altering distance function;

(2)
\(A\cup B\) has a cyclic representation w.r.t. the pair \((f,g)\);

(3)
there exists \(0<\delta<1\) such that the following condition holds:
$$ x \in A, y \in B \quad\mbox{with } x \preceq y \mbox{ or } y \preceq x \quad \Longrightarrow\quad \psi \bigl(s^{3}d(fx,gy) \bigr)\leq\delta\psi \bigl(M_{s}(x,y) \bigr). $$
Definition 4.3
Let \((X,d,\preceq)\) be a complete partially ordered bmetric space with coefficient \(s \geq1\) and A, B be nonempty closed subsets of X with \(X=A\cup B\). Let \(f, g : X \to X\) be two mappings. The pair \((f,g)\) is said to be \((A,B)\) weakly increasing if \(fx \preceq gfx\) for all \(x \in A\) and \(gx \preceq fgx\) for all \(x \in B\).
Definition 4.4
We say that a partially ordered bmetric space \((X,d,\preceq)\) satisfies the property (P) if \(\{x_{n}\}\) being a ⪯nondecreasing sequence in X and \(x_{n} \to x\) as \(n \to\infty\), then \(x_{n} \preceq x\) for all \(n \in\mathbb{N}\).
Theorem 4.5
Let \((X,d,\preceq)\) be a complete partially ordered bmetric space with coefficient \(s \geq1\) and A, B be nonempty closed subsets of X. Suppose that \(f, g : X \to X\) are two mappings such that the pair \((f,g)\) is \((A,B)\)weakly increasing and the following conditions hold:

(1)
the pair \((f,g)\) is a cyclic \((\psi,A,B)_{s}\)contraction;

(2)
f or g is bcontinuous.
Then f and g have a common fixed point in \(A \cap B\).
Proof
Define a mapping \(\alpha: X \times X \rightarrow[0,\infty)\) by
in Theorem 3.3, then we get this result. □
By using the same technique in Theorem 4.5 with Theorem 3.5 and Corollaries 3.6 and 3.7, we get the following result.
Theorem 4.6
Let \((X,d,\preceq)\) be a complete partially ordered bmetric space with coefficient \(s \geq1\) and A, B be nonempty closed subsets of X. Suppose that \(f, g : X \to X\) are two mappings such that the pair \((f,g)\) is \((A,B)\)weakly increasing and the following condition holds:

(1)
the pair \((f,g)\) is a cyclic \((\psi,A,B)_{s}\)contraction;

(2)
X satisfies the property (P).
Then f and g have a common fixed point in \(A \cap B\).
Corollary 4.7
Let \((X,d,\preceq)\) be a complete partially ordered bmetric space with coefficient \(s \geq1\) and A, B be nonempty closed subsets of X with \(X=A\cup B\). Suppose that \(\psi: [0,\infty) \rightarrow[0,\infty)\) and \(f : X \to X\) are two mappings such that \(fx \preceq ffx\) for all \(x \in X\) and the following conditions hold:

(1)
ψ is an altering distance, f is a cyclic mapping and there exists \(0<\delta<1\) such that
$$\begin{aligned} &x \in A, y \in B \quad\textit{with } x \preceq y \textit{ or } y \preceq x \\ &\quad\Longrightarrow\quad \psi \bigl(s^{3}d(fx,fy) \bigr)\leq\delta \psi \biggl(\max \biggl\{ d(x,y),d(x,fx),d(y,fy),\frac{d(x,fy)+d(y,fx)}{2s} \biggr\} \biggr); \end{aligned}$$ 
(2)
f is bcontinuous.
Then f has a fixed point in \(A \cap B\).
Corollary 4.8
Let \((X,d,\preceq)\) be a complete partially ordered bmetric spaces with coefficient \(s \geq1\) and A, B be nonempty closed subsets of X with \(X=A\cup B\). Suppose that \(\psi: [0,\infty) \rightarrow[0,\infty)\) and \(f : X \to X\) are two mappings such that \(fx \preceq ffx\) for all \(x \in X\) and the following conditions hold:

(1)
ψ is altering distance, f is a cyclic mapping, and there exists \(0<\delta<1\) such that
$$\begin{aligned} &x \in A, y \in B \quad\textit{with } x \preceq y \textit{ or } y \preceq x \\ &\quad\Longrightarrow\quad \psi \bigl(s^{3}d(fx,fy) \bigr)\leq\delta \psi \biggl(\max \biggl\{ d(x,y),d(x,fx),d(y,fy),\frac{d(x,fy)+d(y,fx)}{2s} \biggr\} \biggr); \end{aligned}$$ 
(2)
X satisfies the property (P).
Then f has a fixed point in \(A \cap B\).
In 2013, Shatanawi and Postolache [6] introduced the notion of a cyclic \((\psi,A,B)\)contraction pair in partially ordered metric spaces as follows.
Definition 4.9
([6])
Let \((X,d,\preceq)\) be a partially ordered metric space and A, B be two nonempty closed subsets of X. Let \(\psi:[0,\infty) \rightarrow[0,\infty)\) and \(f, g : X \to X\) be three mappings. The pair \((f,g)\) is called a cyclic \((\psi,A,B)\)contraction if

(1)
ψ is an altering distance function;

(2)
\(A\cup B\) has a cyclic representation with respect to the pair \((f,g)\);

(3)
there exists \(0<\delta<1\) such that, for any comparable elements \(x,y \in X\) with \(x \in A\) and \(y \in B\),
$$ \psi \bigl(d(fx,gy) \bigr)\leq\delta\psi \biggl(\max \biggl\{ d(x,y),d(x,fx),d(y,gy), \frac {d(x,gy)+d(y,fx)}{2} \biggr\} \biggr). $$
Since the class of bmetric spaces is effectively larger than that of metric spaces, we can obtain the result of Shatanawi and Postolache [6] from our results.
Corollary 4.10
([6])
Let \((X,d,\preceq)\) be a complete partially ordered metric spaces and A, B be nonempty closed subsets of X. Suppose that \(f, g : X \to X\) are two mappings such that the pair \((f,g)\) is \((A,B)\)weakly increasing and the following conditions hold:

(1)
the pair \((f,g)\) is a cyclic \((\psi,A,B)\)contraction;

(2)
f or g is continuous.
Then f and g have a common fixed point in \(A \cap B\).
Corollary 4.11
([6])
Let \((X,d,\preceq)\) be a complete partially ordered metric space and A, B be nonempty closed subsets of X. Suppose that \(f, g : X \to X\) are two mappings such that the pair \((f,g)\) is \((A,B)\)weakly increasing and the following conditions hold:

(1)
the pair \((f,g)\) is a cyclic \((\psi,A,B)\)contraction;

(2)
X satisfies the property (P).
Then f and g have a common fixed point in \(A \cap B\).
Corollary 4.12
([6])
Let \((X,d,\preceq)\) be a complete partially ordered metric space and A, B be nonempty closed subsets of X with \(X=A\cup B\). Suppose that \(\psi: [0,\infty) \rightarrow[0,\infty)\) and \(f : X \to X\) are two mappings such that \(fx \preceq ffx\) for all \(x \in X\) and the following conditions hold:

(1)
ψ is altering distance, f is a cyclic mapping, and there exists \(0<\delta<1\) such that
$$\begin{aligned}[b] &x \in A, y \in B \quad\textit{with } x \preceq y \textit{ or } y \preceq x \\ &\quad\Longrightarrow\quad \psi \bigl(d(fx,fy) \bigr)\leq\delta\psi \biggl(\max \biggl\{ d(x,y),d(x,fx),d(y,fy),\frac{d(x,fy)+d(y,fx)}{2} \biggr\} \biggr); \end{aligned} $$ 
(2)
f is continuous.
Then f has a fixed point in \(A \cap B\).
Corollary 4.13
([6])
Let \((X,d,\preceq)\) be a complete partially ordered metric spaces and A, B be nonempty closed subsets of X with \(X=A\cup B\). Suppose that \(\psi: [0,\infty) \rightarrow[0,\infty)\) and \(f : X \to X\) are two mappings such that \(fx \preceq ffx\) for all \(x \in X\) and the following conditions hold:

(1)
ψ is altering distance, f is a cyclic mapping and there exists \(0<\delta<1\) such that
$$\begin{aligned} &x \in A, y \in B \quad \textit{with } x \preceq y \textit{ or } y \preceq x \\ &\quad\Longrightarrow\quad \psi \bigl(d(fx,fy) \bigr)\leq\delta\psi \biggl(\max \biggl\{ d(x,y),d(x,fx),d(y,fy),\frac{d(x,fy)+d(y,fx)}{2} \biggr\} \biggr); \end{aligned}$$ 
(2)
X satisfies the property (P).
Then f has a fixed point in \(A \cap B\).
Conclusions
The study of fixed points of mappings and common fixed points of pair of mappings satisfying cyclic contractive conditions has been the focus of vigorous research activity in the last years. As a consequence, many mathematicians obtained more results in this direction. In this paper, the concept of new generalized cyclic contraction pairs in bmetric spaces is introduced. Based on this concept, we have studied the existence of common fixed point results for such pairs in bmetric spaces. Some illustrative examples are furnished which demonstrate the validity of the hypotheses and degree of utility of our results. Also, we can derive some common fixed points existence results for mappings satisfying a generalized cyclic contractive condition in partially ordered bmetric spaces from our main results. These results improve and generalize the main results of Shatanawi and Postolache [6].
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Acknowledgements
This work was support by Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST). 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. The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100).
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Yamaod, O., Sintunavarat, W. & Cho, Y.J. Common fixed point theorems for generalized cyclic contraction pairs in bmetric spaces with applications. Fixed Point Theory Appl 2015, 164 (2015). https://doi.org/10.1186/s136630150409z
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MSC
 47H09
 47H10
Keywords
 bmetric spaces
 common fixed points
 cyclic contraction mappings
 partially ordered set