- Research
- Open Access
Common fixed point theorems for generalized cyclic contraction pairs in b-metric spaces with applications
- Oratai Yamaod^{1},
- Wutiphol Sintunavarat^{1}Email author and
- Yeol Je Cho^{2, 3}Email author
https://doi.org/10.1186/s13663-015-0409-z
© Yamaod et al. 2015
- Received: 21 April 2015
- Accepted: 21 August 2015
- Published: 17 September 2015
Abstract
In this paper, we introduce the notion of generalized cyclic contraction pairs in b-metric 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 b-metric spaces. Our results generalize and improve the result of Shatanawi and Postolache (Fixed Point Theory Appl. 2013:60, 2013) and several well-known results given by some authors in metric and b-metric spaces.
Keywords
- b-metric spaces
- common fixed points
- cyclic contraction mappings
- partially ordered set
MSC
- 47H09
- 47H10
1 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 self-mappings 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 b-metric spaces as a generalization of metric spaces and also proved Banach’s contraction principle in b-metric spaces, which is a generalization of Banach’s contraction principle in metric spaces. Afterward, many mathematicians have studied fixed point results for single-valued and multi-valued mappings in b-metric spaces (see [9–13]).
In this paper, we introduce the concept of new generalized cyclic contraction pairs in b-metric spaces and establish some fixed point theorems for such pairs in the setting of b-metric 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 b-metric spaces. Our result generalizes and improves the corresponding results of Shatanawi and Postolache [6] and several well-known results of fixed point and common fixed point theorems given by some authors in metric and b-metric spaces.
2 Preliminaries
Throughout this paper, we denote by \(\mathbb{N}\), \(\mathbb{R}_{+}\) and \(\mathbb{R}\) the sets of positive integers, non-negative 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
- (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
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])
On the other hand, Bakhtin [8] introduced the concept of a b-metric space as follows.
Definition 2.5
([8])
- (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 \(x,y,z, \in X\).
Every metric space is a b-metric space with \(s=1\) and so the class of b-metric spaces is larger than the class of metric spaces. In general, a b-metric space does not necessarily need to be a metric space.
Now, we give some known examples of a b-metric which show that a b-metric space is a real generalization of metric spaces as follows.
Example 2.6
Example 2.7
Example 2.8
Next, we give the concepts of convergence, a Cauchy sequence, b-continuity, and completeness and closedness in a b-metric space.
Definition 2.9
([10])
- (1)
b-convergent 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 b-Cauchy sequence if \(d(x_{n}, x_{m}) \rightarrow0\) as \(n,m \rightarrow\infty\).
Proposition 2.10
([10])
- (1)
a b-convergent sequence has a unique limit;
- (2)
each b-convergent sequence is a b-Cauchy sequence;
- (3)
in general, a b-metric is not continuous.
We need the following lemma as regards b-convergent sequences in the proof of our results.
Lemma 2.11
([14])
Definition 2.12
([10])
- (1)
The space \((X, d)\) is b-complete if every b-Cauchy sequence in X b-converges;
- (2)
a function \(f : X\rightarrow X'\) is b-continuous at a point \(x\in X\) if it is b-sequentially continuous at x, that is, whenever \(\{x_{n}\}\) is b-convergent to x, \(\{fx_{n}\}\) is b-convergent to fx.
Definition 2.13
([10])
Definition 2.14
([10])
Let \((X, d)\) be a b-metric space. Then a subset \(Y \subseteq X\) is said to be closed if and only if, for each sequence \(\{x_{n}\}\) in Y which b-converges 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.
3 Main results
Definition 3.1
- (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 b-metric 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
- (1)
the pair \((f,g)\) is a cyclic α-\((\psi,A,B)_{s}\)-contraction;
- (2)
f or g is b-continuous;
- (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 \(\alpha(z,z)\geq1\).
Proof
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 \(x_{2n} = x_{2n+1}\).
- (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)
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 \(z=fz\).
Theorem 3.3 can be proved without assuming the b-continuity of f or the b-continuity of g. For this instance, we assume that X satisfies the following property.
Definition 3.4
Let \((X,d)\) be a b-metric 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
- (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 \(\alpha(z,z)\geq1\).
Proof
Taking \(f=g\) in Theorems 3.3 and 3.5, we have the following result.
Corollary 3.6
- (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 b-continuous;
- (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 \(\alpha(z,z)\geq1\).
Corollary 3.7
- (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 \(\alpha(z,z)\geq1\).
Now, we give an example to illustrate the utility of Theorem 3.5.
Example 3.8
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.
- (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 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.
4 Some particular cases
In this section, we give some fixed point results on partially ordered b-metric 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 b-metric space if \((X,d)\) is a b-metric space and \((X,\preceq)\) is a partially ordered set.
Definition 4.2
- (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 b-metric 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 b-metric 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
- (1)
the pair \((f,g)\) is a cyclic \((\psi,A,B)_{s}\)-contraction;
- (2)
f or g is b-continuous.
Proof
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
- (1)
the pair \((f,g)\) is a cyclic \((\psi,A,B)_{s}\)-contraction;
- (2)
X satisfies the property (P).
Corollary 4.7
- (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 b-continuous.
Corollary 4.8
- (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).
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])
- (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 b-metric 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])
- (1)
the pair \((f,g)\) is a cyclic \((\psi,A,B)\)-contraction;
- (2)
f or g is continuous.
Corollary 4.11
([6])
- (1)
the pair \((f,g)\) is a cyclic \((\psi,A,B)\)-contraction;
- (2)
X satisfies the property (P).
Corollary 4.12
([6])
- (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.
Corollary 4.13
([6])
- (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).
5 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 b-metric spaces is introduced. Based on this concept, we have studied the existence of common fixed point results for such pairs in b-metric 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 b-metric spaces from our main results. These results improve and generalize the main results of Shatanawi and Postolache [6].
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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