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Common fixed point theorems for Geraghty’s type contraction mappings using the monotone property with two metrics
 Juan MartínezMoreno^{1},
 Wutiphol Sintunavarat^{2}Email author and
 Yeol Je Cho^{3, 4}Email author
https://doi.org/10.1186/s136630150426y
© MartínezMoreno et al. 2015
Received: 21 April 2015
Accepted: 14 September 2015
Published: 26 September 2015
Abstract
The main aim of this paper is to obtain some new common fixed point theorems for Geraghty’s type contraction mappings using the monotone property with two metrics and to give some examples to illustrate the main results. Further, by using our main results, we prove some results about multidimensional common fixed points. Our results generalize and extend some recent results given by Kadelburg et al. (Fixed Point Theory Appl. 2015:27, 2015) and Choudhurya and Kundu (J. Nonlinear Sci. Appl. 5:259270, 2012).
Keywords
 coincidence point
 common fixed point
 Geraghty’s type contraction
 mixed monotone property
 partially ordered set
1 Introduction and preliminaries
Theorem 1.1
([1])
Recently, AminiHarandi and Emami [2] extended this result to the setting of partially ordered metric spaces as follows:
Theorem 1.2
([2])
 (1)
f is continuous or
 (2)
for any nondecreasing sequence \(\{x_{n}\}\) in X, if \(x_{n}\to x\in X\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all \(n\geq1\).
If, moreover, for all \(x,y\in X\), there exists \(z\in X\) comparable with x and y, then the fixed point of f is unique.
For more generalizations of Theorems 1.1 and 1.2, see [3–5].
On the other hand, several authors have studied fixed point theory in spaces equipped with two metrics (see [6–8]). Especially, Agarwal and O’Regan [6] proved the following:
Theorem 1.3
([6])
 (1)
\(d(x_{0}, Fx_{0})<(1q)r\);
 (2)
if \(d\ngeq d'\), assume that F is uniformly continuous from \((B(x_{0}, r),d)\) into \((X, d')\);
 (3)
if \(d\neq d'\), assume that F is continuous from \((\overline {B(x_{0}, r)^{d'}}, d)\) into \((X, d')\).
The aim of this paper is to study some new common fixed point theorems for Geraghty’s type contraction mappings using the monotone property with two metrics, which is an important advantage to compare with well known fixed point theorems in metric spaces. Further, we give some examples to illustrate the main results. The main results in this paper generalize, unify, and extend some recent results given by some authors.
2 Main results
In this section, we prove some fixed point results for generalized contractions on spaces with two metrics.
Throughout this paper, \((X,\preceq)\) denotes a partially ordered set. By \(x\succeq y\), we mean \(y\preceq x\). Let \(f,g:X\rightarrow X\) be two mappings. A mapping f is said to be gnondecreasing (resp., gnonincreasing) if, for all \(x,y\in X\), \(gx\preceq gy\) implies \(fx \preceq fy\) (resp., \(fy \preceq fx\)). If g is the identity mapping, then f is said to be nondecreasing (resp., nonincreasing). Let \(d'\), d be two metrics on X. By \(d < d'\) (resp., \(d\leq d'\)), we mean \(d(x,y) < d'(x,y)\) (resp., \(d(x,y) \leq d'(x,y)\)) for all \(x,y\in X\).
Also, we give some essential concepts which are useful for our main results:
Definition 2.1
([9])
Definition 2.2
([9])
Let \((X,d)\) and \((Y,d')\) be two metric spaces and \(f:X \rightarrow Y\) and \(g:X\rightarrow X\) be two mappings. A mapping f is said to be guniformly continuous on X if, for any real number \(\epsilon> 0\), there exists \(\delta> 0\) such that \(d'(fx,fy) < \epsilon\) whenever \(x, y \in X\) and \(d(gx, gy) < \delta\). If g is the identity mapping, then f is said to be uniformly continuous on X.
Now, we give the main result in this paper.
Theorem 2.3
 (1)
\(g:(X,d')\to(X,d')\) is continuous and \(g(X)\) is \(d'\)closed;
 (2)
\(f(X)\subseteq g(X)\);
 (3)
there exists \(x_{0}\in X\) such that \(gx_{0}\preceq fx_{0}\);
 (4)there exists \(\theta\in\Theta\) such thatfor all \(x,y\in X\) with \(gx\preceq gy\) or \(gx\succeq gy\):$$ d(fx,fy)\leq\theta\bigl(d(gx,gy)\bigr)d(gx,gy) $$(2.1)
 (5)
if \(d\ngeq d'\), assume that \(f:(X,d)\to(X,d')\) is guniformly continuous;
 (6)
if \(d\neq d'\), assume that \(f:(X,d')\to(X,d')\) is continuous and g and f are \(d'\)compatible;
 (7)
if \(d= d'\), assume that (a) f is continuous and g and f are compatible or (b) for any nondecreasing sequence \(\{x_{n}\}\) in X, if \(x_{n}\to x\in X\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all \(n\geq1\).
Proof
Finally, we prove that u is a common fixed point of f and g. We consider two cases:
Case I: \(d \neq d'\).
Case II: \(d= d'\).
Now, we give some examples to illustrate Theorem 2.3.
Example 2.4
Next, we show that the conditions (1)(7) in Theorem 2.3 hold as follows:
(1) We can easily check that \(g:(X,d') \rightarrow(X,d')\) is continuous. Also, we can see that \(g(X)=[0,\infty)\) is \(d'\)closed.
(2) By the definition of f and g, we can see that \(f(X) = g(X)\).
(3) It is easy to see that there exists a point \(x_{0} \in X\) such that \(gx_{0} \preceq fx_{0}\).
(7) Since \(d\neq d'\), we have nothing to do to show this condition.
Consequently, all the conditions of Theorem 2.3 hold. Therefore, g and f have a coincidence point and, further, a point 0 is a coincidence point of the mappings g and f.
Example 2.5
Next, we show that the conditions (1)(7) in Theorem 2.3 hold as follows:
(1) We can easily check that \(g:(X,d') \rightarrow(X,d')\) is continuous. Also, we can see that \(g(X)=[0,\infty)\) is \(d'\)closed.
(2) By the definition of f and g, we can see that \(f(X) = g(X)\).
(3) It is easy to see that there exists a point \(x_{0} \in X\) such that \(gx_{0} \preceq fx_{0}\).
(5) It follows from \(d \geq d'\) that we have nothing to do to show this condition.
(7) Since \(d\neq d'\), we have nothing to do to show this condition.
Consequently, all the conditions of Theorem 2.3 hold. Therefore, g and f have a coincidence point and, further, the points 0 and 1 are coincidence points of the mappings g and f.
Putting \(g=I_{X}\), where \(I_{X}\) is the identity mapping on X in Theorem 2.3, we obtain the following:
Corollary 2.6
 (1)
there exists \(x_{0}\in X\) such that \(x_{0}\preceq fx_{0}\);
 (2)there exists \(\theta\in\Theta\) such thatfor all \(x,y\in X\) with \(x\preceq y\) or \(x\succeq y\);$$ d(fx,fy)\leq\theta\bigl(d(x,y)\bigr)d(x,y) $$(2.8)
 (3)
if \(d\ngeq d'\), assume that \(f:(X,d)\to(X,d')\) is uniformly continuous;
 (4)
if \(d\neq d'\), assume that \(f:(X,d)\to(X,d)\) is continuous;
 (5)
if \(d= d'\), then (a) f is continuous or (b) for any nondecreasing sequence \(\{x_{n}\}\) in X, if \(x_{n}\to x\in X\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all n.
Taking \(d=d'\) in Theorem 2.3, we have the following:
Theorem 2.7
 (1)
g is continuous and \(g(X)\) is closed;
 (2)
\(f(X)\subseteq g(X)\);
 (3)
there exists \(x_{0}\in X\) such that \(gx_{0}\preceq fx_{0}\);
 (4)there exists \(\theta\in\Theta\) such thatfor all \(x,y\in X\) with \(gx\preceq gy\) or \(gx\succeq gy\);$$ d(fx,fy)\leq\theta\bigl(d(gx,gy)\bigr)d(gx,gy) $$(2.9)
 (5)
(a) f is continuous and g and f are compatible or (b) for any nondecreasing sequence \(\{x_{n}\}\) in X, if \(x_{n}\to x\in X\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all n.
Theorem 2.8
 (8)
for any \(x,u\in X\), there exists \(y\in X\) such that fy is comparable to both fx and fu.
Proof
Theorem 2.3 implies that there exists a coincidence point \(x\in X\), that is, \(gx=fx\). Suppose that there exists another coincidence point \(u\in X\) and hence \(gu=fu\).
3 Some particular cases
First, we give some definitions for the main results in this section.
Definition 3.1
If, in the previous relations, g is the identity mapping, then F is said to have the monotone property.
Definition 3.2
 (C_{1}):

a coupled fixed point of F if \(x=F(x,y)\) and \(y=F(y,x)\);
 (C_{2}):

a coupled coincidence point of g and F if \(gx=F(x,y)\) and \(gy=F(y,x)\) and, in this case, a point \((gx,gy)\) is called a coupled point of coincidence;
 (C_{3}):

a common coupled fixed point of g and F if \(x=gx=F(x,y)\) and \(y=gy=F(y,x)\).
Definition 3.3
([12])
Now, we prove some results to show how coupled notions (as the compatibility) can be reduced to the unidimensional case using the mappings defined as follows:
For instance, the following lemma guarantees that the 2dimensional notion of common fixed coincidence points can be interpreted in terms of two mappings \(T_{F}^{2}\) and \(G^{2}\).
Lemma 3.4
 (1)
a coupled fixed point of F if and only if it is a fixed point of the mapping \(T_{F}^{2}\);
 (2)
a coupled coincidence point of F and g if and only if it is a coincidence point of two mappings \(T_{F}^{2}\) and \(G^{2}\);
 (3)
a coupled fixed point of F and g if and only if it is a common fixed point of two mappings \(T_{F}^{2}\) and \(G^{2}\).
Proof
Now, we show how to use Theorem 2.3 in order to deduce coupled fixed point results.
Theorem 3.5
 (1)
\(g:(X,d')\to(X,d')\) is continuous and \(g(X)\) is \(d'\)closed;
 (2)
\(F(X\times X)\subseteq g(X)\);
 (3)
there exist \(x_{0},y_{0}\in X\) such that \(gx_{0}\preceq F(x_{0},y_{0})\) and \(gy_{0}\preceq F(y_{0},x_{0})\);
 (4)there exists \(\theta\in\Theta\) such thatfor all \(x,y,u,v\in X\) with \(gx\preceq gu\) and \(gy\preceq gv\) or \(gx\succeq gu\) and \(gy\succeq gv\);$$ d\bigl(F(x,y),F(u,v)\bigr)\leq\theta\bigl(\max\bigl\{ d(gx,gu),d(gy,gv)\bigr\} \bigr)\max\bigl\{ d(gx,gu),d(gy,gv)\bigr\} $$(3.3)
 (5)
if \(d\ngeq d'\), assume that \(F:(X,d)\times(X,d)\to(X,d')\) is guniformly continuous;
 (6)
if \(d\neq d'\), assume that \(F:(X,d')\times(X,d')\to (X,d')\) is continuous and g and F are \(d'\)compatible;
 (7)
if \(d= d'\), assume that (a) F is continuous and g and F are compatible or (b) for any nondecreasing sequence \(\{x_{n}\}\) in X, if \(x_{n}\to x\in X\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all \(n\geq1\).
Proof
Taking \(d=d'\) in Theorem 3.5, we get the following result in [13]:
Corollary 3.6
([13], Theorem 3.1)
 (1)
g is continuous and \(g(X)\) is closed;
 (2)
\(F(X\times X)\subseteq g(X)\);
 (3)
there exist \(x_{0},y_{0}\in X\) such that \(gx_{0}\preceq F(x_{0},y_{0})\) and \(gy_{0}\preceq F(y_{0},x_{0})\);
 (4)there exists \(\theta\in\Theta\) such thatfor all \(x,y,u,v\in X\) with \(gx\preceq gu\) and \(gy\preceq gv\) or \(gx\succeq gu\) and \(gy\succeq gv\);$$ d\bigl(F(x,y),F(u,v)\bigr)\leq\theta\bigl(\max\bigl\{ d(gx,gu),d(gy,gv)\bigr\} \bigr)\max\bigl\{ d(gx,gu),d(gy,gv)\bigr\} $$(3.4)
 (5)
(a) F is continuous and g and F are compatible or (b) for any nondecreasing sequence \(\{x_{n}\}\) in X, if \(x_{n}\to x\in X\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all \(n\geq1\).
Definition 3.7
If, in the previous relations, g is the identity mapping, then F is said to have the mixed monotone property.
Now, we show how to use Theorem 2.3 in order to deduce coupled fixed point results with the gmixed monotone properties.
Theorem 3.8
 (1)
\(g:(X,d')\to(X,d')\) is continuous and \(g(X)\) is \(d'\)closed;
 (2)
\(F(X\times X)\subseteq g(X)\);
 (3)
there exist \(x_{0},y_{0}\in X\) such that \(gx_{0}\preceq F(x_{0},y_{0})\) and \(gy_{0}\succeq F(y_{0},x_{0})\);
 (4)there exists \(\theta\in\Theta\) such thatfor all \(x,y,u,v\in X\) satisfying \(gx\preceq gu\) and \(gy\succeq gv\) or \(gx\succeq gu\) and \(gy\succeq gv\);$$ d\bigl(F(x,y),F(u,v)\bigr)\leq\theta \biggl(\frac{d(gx,gu)+d(gy,gv)}{2} \biggr) \frac {d(gx,gu)+d(gy,gv)}{2} $$(3.5)
 (5)
if \(d\ngeq d'\), assume that \(F:(X,d)\times(X,d)\to(X,d')\) is guniformly continuous;
 (6)
if \(d\neq d'\), assume that \(F:(X,d')\times(X,d')\to (X,d')\) is continuous and g and F are \(d'\)compatible;
 (7)
if \(d= d'\), assume that (a) F is continuous and g and F are compatible or (b1) for any nondecreasing sequence \(\{x_{n}\}\) in X, if \(x_{n}\to x\in X\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all \(n \in\mathbb{N}\) and (b2) for any nonincreasing sequence \(\{x_{n}\}\) in X, if \(x_{n}\to x\in X\) as \(n\to\infty\), then \(x_{n}\succeq x\) for all \(n \in\mathbb{N}\).
Proof
Remark 3.9
In the above result, if g is the identity mapping and \(d=d'\), then we obtain Theorem 2.1 in [14].
Declarations
Acknowledgements
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. Also, Yeol Je Cho was supported by 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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