Common fixed point theorems for hybrid contractive pairs with the \((\mathit{CLR})\)-property
- Afrah AN Abdou^{1}Email author
https://doi.org/10.1186/s13663-015-0378-2
© Abdou 2015
Received: 21 April 2015
Accepted: 6 July 2015
Published: 7 August 2015
Abstract
In this work, we introduce the \((\mathit{CLR})\)-property for the hybrid pairs of single-valued and multi-valued mappings and give some coincidence and common fixed point theorems for the hybrid pairs of some contractive conditions. Also, we will give some examples to illustrate the main results in this paper. Our results extend and improve some results given by some authors.
Keywords
MSC
1 Introduction
In 1969, Nadler [1] introduced the notion of a multi-valued (set-valued) contractive mapping in a metric space and also proved Banach’s fixed point theorem for a multi-valued mapping in a metric space. Since Nadler, many authors have studied Banach’s fixed point theorem for multi-valued mappings in several ways [2–7].
Especially, fixed point theorems for the hybrid contractive pairs of single-valued and multi-valued mappings are always be an interesting area of research due to its majority on only single-valued contractive mappings or only multi-valued contractive mappings in general spaces [8]. Besides, there are many results as regards fixed point theorems for multi-valued mappings in metric spaces with different contractive conditions and applications. For more details, we refer to [9–11] and references therein.
In 1982, Sessa [12] first studied common fixed points results for weakly commuting pair of single-valued mappings in metric spaces. Afterward, Jungck [13] introduced the concept of compatible single-valued mappings in order to generalize the concept of weak commutativity by Sessa [12] and showed that weakly commuting mappings are compatible, but the converse is not true. In 1996, Jungck [14] introduced the concept of weakly compatibility for single-valued mappings. Afterward, Aamri and El Moutawakil [15] introduced the notion of the property \((E.A.)\), which is a special case of the tangential property due to Sastry and Krishna Murthy [16]. In 2011, Sintunaravat and Kumam [17] showed that the notion of the property \((E.A.)\) always requires the completeness (or closedness) of the underlying subspaces for the existence of common fixed points for single-valued mappings. Hence they coined the idea of common limit in the range (for brevity, called the \((\mathit{CLR})\)-property), which relaxes the requirement of completeness (or closedness) of the underlying subspace. They also proved common fixed point results for single-valued mappings via this concept in fuzzy metric spaces. For more details on the \((\mathit{CLR})\)-property, refer to [18–20] and therein.
Inspired by the notion of the property \((\mathit{CLR})\)-property, we introduce the \((\mathit{CLR}_{g})\)-property for the hybrid pairs of single-valued and multi-valued mappings in metric spaces and give some new coincidence and common fixed point theorems under the hybrid pairs satisfying some contractive conditions. Also, we give some examples to illustrate the main results in this paper. Our results improve, extend, and generalize the corresponding results given by some authors.
2 Preliminaries
We also denote by \(\operatorname{Fix}(T)\) the set of all fixed points of a multi-valued mapping T.
Definition 2.1
([13])
In 1989, Kaneko and Sessa [22] introduced the notion of compatible for single-valued and multi-valued mappings as follows.
Definition 2.2
([22])
Remark 2.3
Definition 2.4
([14])
Let \((X,d)\) be a metric space. Two mappings \(f:X\rightarrow X\) and \(T:X\rightarrow \operatorname{CB}(X)\) are said to be weakly compatible if they commute at their coincidence points, i.e., if \(fTx=Tfx\) whenever \(fx\in Tx\).
It is easy to see that two compatible mappings are weakly compatible, but the converse is not true.
Definition 2.5
([17])
Example 2.6
3 Main results
Now, we define the \((\mathit{CLR}_{f})\)-property for a hybrid pairs of single-valued and multi-valued mappings in metric spaces.
Definition 3.1
Remark 3.2
Note that, if \(f(X)\) is closed, then a noncompatible hybrid pair \((f,T)\) satisfies the \((\mathit{CLR}_{f})\) w.r.t. T.
Now, we give an example for two mappings satisfying the \((\mathit{CLR}_{f})\)-property w.r.t. T.
Example 3.3
Here, we state and prove the main result in this paper.
Theorem 3.4
- (1)
f and T satisfy the \((\mathit{CLR}_{f})\)-property w.r.t. T;
- (2)for all \(x,y\in X\),where \(p\geq1\) and \(\varphi:[0,\infty)\rightarrow[0,\infty)\) is a continuous monotone increasing function such that \(\varphi(0)=0\) and \(\varphi(t)< t\) for all \(t>0\).$$\begin{aligned} H^{p}(Tx,Ty) \leq&\varphi \biggl(\max \biggl\{ d^{p}(fx,fy), \frac {d^{p}(fx,Tx)d^{p}(fy,Ty)}{1+d^{p}(fx,fy)}, \\ & \frac{d^{p}(fx,Ty)d^{p}(fy,Tx)}{1+d^{p}(fx,fy)} \biggr\} \biggr), \end{aligned}$$
Proof
From Remark 3.2, we have the following result.
Corollary 3.5
- (1)
f and T are noncompatible;
- (2)for all \(x,y\in X\),where \(p\geq1\) and \(\varphi:[0,\infty)\rightarrow[0,\infty)\) is a continuous monotone increasing function such that \(\varphi(0)=0\) and \(\varphi(t)< t\) for all \(t>0\).$$\begin{aligned} H^{p}(Tx,Ty) \leq&\varphi \biggl(\max \biggl\{ d^{p}(fx,fy), \frac {d^{p}(fx,Tx)d^{p}(fy,Ty)}{1+d^{p}(fx,fy)}, \\ & \frac{d^{p}(fx,Ty)d^{p}(fy,Tx)}{1+d^{p}(fx,fy)} \biggr\} \biggr), \end{aligned}$$
Theorem 3.6
Let \((X,d)\) be a metric space and let \(f:X\rightarrow X\) and \(T:X\rightarrow \operatorname{CB}(X)\) are two mappings satisfying the conditions (1), (2) of Theorem 3.4. If f and T are weakly compatible at a and \(ffa=fa\) for some \(a\in C(f,T) \neq\emptyset\), then f and T have a common fixed point in X.
Proof
Corollary 3.7
Let \((X,d)\) be a metric space and let \(f:X\rightarrow X\), \(T:X\rightarrow \operatorname{CB}(X)\) be two mappings satisfying the conditions (1), (2) of Theorem 3.4. If f and T are weakly compatible at a and \(ffa=fa\) for all \(a\in C(f,T) \neq\emptyset\), then f and T have a common fixed point.
Next, we give one interesting example to illustrate Theorems 3.4 and 3.6, but Corollary 3.7 is not applicable.
Example 3.8
Next, we claim that f and T have a common fixed point in X by using Theorem 3.6. Also, we can see that f and T are weakly compatible at a point a and \(ffa=fa\) for \(a=1 \in C(f,T)\). So, all the conditions of Theorem 3.6 are satisfied. Therefore, f and T have a common fixed point in X. In this case, the point 1 is a unique common fixed point of f and T.
Remark 3.9
From Example 3.8, we can see that f and T are not weakly compatible at a point a with \(a\in C(f,T) = [1,\frac{1+\sqrt{5}}{2} ]\). Also, \(ffa\neq fa\) for all \(a\in C(f,T) = [1,\frac{1+\sqrt {5}}{2} ]\). Therefore, Corollary 3.7 cannot be applicable in this case.
If we take \(p=1\) in Theorem 3.6, then we have the following result.
Corollary 3.10
- (1)
f and T satisfy the \((\mathit{CLR}_{f})\)-property w.r.t. T;
- (2)for all \(x,y\in X\),where \(p\geq1\) and \(\varphi:[0,\infty)\rightarrow[0,\infty)\) is a continuous monotone increasing function such that \(\varphi(0)=0\) and \(\varphi(t)< t\) for all \(t>0\).$$ H(Tx,Ty) \leq\varphi \biggl(\max \biggl\{ d(fx,fy), \frac {d(fx,Tx)d(fy,Ty)}{1+d(fx,fy)}, \frac{d(fx,Ty)d(fy,Tx)}{1+d(fx,fy)} \biggr\} \biggr), $$
If f and T are weakly compatible at a point a and \(ffa=fa\) for some \(a\in C(f,T) \neq\emptyset\), then f and T have a common fixed point in X.
Corollary 3.11
- (1)
f and T satisfy the \((\mathit{CLR}_{f})\)-property w.r.t. T;
- (2)for all \(x,y\in X\),where \(p\geq1\) and \(\varphi:[0,\infty)\rightarrow[0,\infty)\) is a continuous monotone increasing function such that \(\varphi(0)=0\) and \(\varphi(t)< t\) for all \(t>0\).$$ H(Tx,Ty) \leq \varphi \biggl(\max \biggl\{ d(fx,fy), \frac {d(fx,Tx)d(fy,Ty)}{1+d(fx,fy)}, \frac{d(fx,Ty)d(fy,Tx)}{1+d(fx,fy)} \biggr\} \biggr), $$
If f and T are weakly compatible at a point a and \(ffa=fa\) for all \(a\in C(f,T) \neq\emptyset\), then f and T have a common fixed point in X.
Corollary 3.12
- (1)
f and T satisfy the \((\mathit{CLR}_{f})\)-property w.r.t. T;
- (2)there exists \(k\in[0,1)\) such thatfor all \(x,y\in X\).$$ H(Tx,Ty) \leq k\max \biggl\{ d(fx,fy), \frac {d(fx,Tx)d(fy,Ty)}{1+d(fx,fy)}, \frac{d(fx,Ty)d(fy,Tx)}{1+d(fx,fy)} \biggr\} $$
If f and T are weakly compatible at a point a and \(ffa=fa\) for some \(a\in C(f,T) \neq\emptyset\), then f and T have a common fixed point in X.
Proof
Take \(\varphi(t)=kt\) in Corollary 3.10. Then we have the conclusion. □
Corollary 3.13
- (1)
f and T satisfy the \((\mathit{CLR}_{f})\)-property w.r.t. T;
- (2)there exists \(k\in[0,1)\) such thatfor all \(x,y\in X\).$$ H(Tx,Ty) \leq k\max \biggl\{ d(fx,fy), \frac {d(fx,Tx)d(fy,Ty)}{1+d(fx,fy)}, \frac{d(fx,Ty)d(fy,Tx)}{1+d(fx,fy)} \biggr\} $$
If f and T are weakly compatible at a and \(ffa=fa\) for all \(a\in C(f,T) \neq\emptyset\), then f and T have a common fixed point in X.
If we take \(f=I\) (the identity mapping in X) in Theorem 3.6, then we have the following result.
Corollary 3.14
- (1)there exists a sequence \(\{x_{n}\}\) in X such thatfor some \(A\in \operatorname{CB}(X)\) and$$\lim_{n\to\infty}Tx_{n}=A $$for some \(u\in X\);$$\lim_{n\to\infty}x_{n} = u \in A $$
- (2)for all \(x,y\in X\),where \(p\geq1\) and \(\varphi:[0,\infty)\rightarrow[0,\infty)\) is a monotone increasing function such that \(\varphi(0)=0\) and \(\varphi(t)< t\) for all \(t>0\).$$ H^{p}(Tx,Ty) \leq \varphi \biggl(\max \biggl\{ d^{p}(x,y), \frac {d^{p}(x,Tx)d^{p}(y,Ty)}{1+d^{p}(x,y)}, \frac{d^{p}(x,Ty)d^{p}(y,Tx)}{1+d^{p}(x,y)} \biggr\} \biggr), $$
Corollary 3.15
- (1)there exists a sequence \(\{x_{n}\}\) in X such thatfor some \(A\in \operatorname{CB}(X)\) and$$\lim_{n\to\infty}Tx_{n}=A $$for some \(u\in X\);$$\lim_{n\to\infty}x_{n} = u \in A $$
- (2)there exists \(k\in[0,1)\) such thatfor all \(x,y\in X\).$$H(Tx,Ty) \leq k\max \biggl\{ d(x,y), \frac{d(x,Tx)d(y,Ty)}{1+d(x,y)},\frac {d(x,Ty)d(y,Tx)}{1+d(x,y)} \biggr\} $$
Proof
Take \(p=1\) and \(\varphi(t) = kt\) in Corollary 3.14. Then we have the conclusion. □
4 Conclusion
Recently, some authors have required some conditions, that is, the completeness of X, the closedness or the convexity of some suitable subset of X, the continuity of one mapping or more mappings, and the containment of the range of the given mappings, to prove some common fixed point results for single-valued and multi-valued mappings in a metric space X, but, as in our results, if we use the \((\mathit{CLR})\)-property for single-valued and multi-valued mappings, then we do not need the conditions mentioned above.
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support. Also, the author would like to thank Prof. YJ Cho for fruitful discussion and Dr. W Sintunavarat for sending some important papers for the improvement of this paper.
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.
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