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Common coupled fixed point theorems for Geraghtytype contraction mappings using monotone property
 Zoran Kadelburg^{1},
 Poom Kumam^{2, 3}Email author,
 Stojan Radenović^{4} and
 Wutiphol Sintunavarat^{5}Email author
https://doi.org/10.1186/s1366301502785
© Kadelburg et al.; licensee Springer. 2015
 Received: 15 November 2014
 Accepted: 2 February 2015
 Published: 22 February 2015
Abstract
In this paper, we prove common coupled fixed point theorems under a Geraghtytype condition using monotone instead of the often used mixed monotone property. Also we give some sufficient conditions for the uniqueness of a common coupled fixed point. An example illustrating our results is provided.
Keywords
 partially ordered metric space
 coupled fixed point
 mixed monotone property
 Geraghtytype condition
MSC
 47H10
 54H25
1 Introduction
As the Banach contraction principle (BCP) is a power tool for solving many problems in applied mathematics and sciences, it has been improved and extended in many ways. In particular, Geraghty proved in 1973 [1] an interesting generalization of BCP which has had a lot of applications.
The notion of a coupled fixed point was firstly introduced and studied by Opoitsev [2, 3] and then by Guo and Lakshmikantham in [4]. In 2006, Bhaskar and Lakshmikantham [5] were the first to introduce the notion of mixed monotone property. They also studied and proved the following classical coupled fixed point theorems for mappings by using this property under contractive type conditions.
Theorem 1.1
([5])
Theorem 1.2
([5])
 1.
if \(\{x_{n}\}\) is a nondecreasing sequence with \(\{x_{n}\}\rightarrow x\), then \(x_{n}\preceq x\) for all \(n\geq1\),
 2.
if \(\{y_{n}\}\) is a nonincreasing sequence with \(\{y_{n}\}\rightarrow y\), then \(y_{n}\succeq y\) for all \(n\geq1\).
Due to the important role of Theorems 1.1 and 1.2 for the investigation of solutions of nonlinear differential and integral equations, several authors have studied various generalizations of these results (see, e.g., papers [6–18] and the references cited therein). Almost all of them used the mixed monotone property or, in the case of additional mapping \(g:X\to X\), the socalled gmixed monotone property.
Recently, in [19–21], the author established common coupled fixed point theorems by using (g)monotone property instead of (g)mixed monotone property. These kinds of results can be applied in another type of situations, so they give an opportunity to widen the field of applications. In particular the socalled tripled fixed point results (and, more generally, ntupled results) can be more easily handled using monotone property instead of mixed monotone property (see, e.g., [22–24]).
The aim of this work is to prove some common coupled fixed point theorems for Geraghtytype contraction mappings by using monotone and gmonotone property instead of mixed monotone and gmixed monotone property. An illustrative example is presented in this work showing how our results can be used in proving the existence of a common coupled fixed point, while the results of many other papers cannot.
2 Preliminaries
In this section, we give some definitions that are useful for our main results in this paper. 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 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 an identity mapping, then f is said to be nondecreasing (resp., nonincreasing).
Definition 2.1
Definition 2.2
 (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 mappings g and F ifand in this case \((gx,gy)\) is called a coupled point of coincidence;$$ gx=F(x,y) \quad \mbox{and}\quad gy=F(y,x), $$
 (C_{3}):

a common coupled fixed point of mappings g and F if$$ x=gx=F(x,y) \quad \mbox{and}\quad y=gy=F(y,x). $$
Definition 2.3
([25])
3 Main results
 (\(\theta_{1}\)):

\(\theta(s,t)=\theta(t,s)\) for all \(s,t\in[0,\infty)\);
 (\(\theta_{2}\)):

for any two sequences \(\{s_{n}\}\) and \(\{ t_{n}\} \) of nonnegative real numbers,$$ \theta(s_{n},t_{n})\rightarrow1\quad \Rightarrow \quad s_{n},t_{n}\rightarrow0. $$
 (1)
\(\theta(s,t)=k\) for \(s,t\in[0,\infty )\), where \(k\in[0,1)\).
 (2)
\(\theta(s,t)= \bigl\{\scriptsize{\begin{array}{l@{\quad}l} \frac{\ln(1+ks+lt)}{ks+lt}, & s>0\mbox{ or }t>0, \\[1pt] r\in[0,1), & s=0,t=0, \end{array}} \bigr.\) where \(k,l\in(0,1)\).
 (3)
\(\theta(s,t)= \bigl\{\scriptsize{\begin{array}{l@{\quad}l} \frac{\ln(1+\max\{s,t\})}{\max\{s,t\}}, & s>0\mbox{ or }t>0, \\[1pt] r\in[0,1), & s=0,t=0. \end{array}}\bigr. \)
Now, we will prove our main result.
Theorem 3.1
 (i)
g is continuous and \(g(X)\) is closed;
 (ii)
\(F(X\times X)\subset g(X)\) and g and F are compatible;
 (iii)
there exist \(x_{0},y_{0}\in X\) such that \(gx_{0}\preceq F(x_{0},y_{0})\) and \(gy_{0}\preceq F(y_{0},x_{0})\);
 (iv)there exists \(\theta\in\Theta\) such that for all \(x,y,u,v\in X\) satisfying \(gx\preceq gu\) and \(gy\preceq gv\) or \(gx\succeq gu\) and \(gy\succeq gv\),holds true;$$ d\bigl(F(x,y),F(u,v)\bigr)\leq\theta\bigl(d(gx,gu),d(gy,gv)\bigr)\max\bigl\{ d(gx,gu),d(gy,gv)\bigr\} $$(3.1)
 (v)
(a) F is continuous or (b) if, for an increasing sequence \(\{x_{n}\}\) in X, \(x_{n}\to x\in X\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all \(n\in\mathbb{N}\).
Then there exist \(u,v\in X\) such that \(gu=F(u,v)\) and \(gv=F(v,u)\), i.e., g and F have a coupled coincidence point.
Proof
Note that in this case continuity and compatibility assumptions were not needed in the proof. □
Remark 3.2
In Theorem 3.1, the condition that F has the gmonotone property is a substitution for the gmixed monotone property that was used in most of the coupled fixed point results so far. Note that this condition is maybe more natural than the mixed gmonotone property and can be used in various examples.
Putting \(g=I_{X}\), where \(I_{X}\) is an identity mapping on X in Theorem 3.1, we obtain the following.
Corollary 3.3
 (i)
there exist \(x_{0},y_{0}\in X\) such that \(x_{0}\preceq F(x_{0},y_{0})\) and \(y_{0}\preceq F(y_{0},x_{0})\);
 (ii)there exists \(\theta\in\Theta\) such that for all \(x,y,u,v\in X\) satisfying (\(x\preceq u\) and \(y\preceq v\)) or (\(u\preceq x\) and \(v\preceq y\)),holds true;$$ d\bigl(F(x,y),F(u,v)\bigr)\leq\theta\bigl(d(x,u),d(y,v)\bigr)\max\bigl\{ d(x,u),d(y,v)\bigr\} $$
 (iii)
(a) F is continuous or (b) if \(\{x_{n}\}\) is an increasing sequence in X and \(x_{n}\to x\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all n.
Taking \(\theta(t_{1},t_{2})=k\) with \(k\in[0,1)\) for all \(t_{1},t_{2}\in [0,\infty)\) in Theorem 3.1 and Corollary 3.3, we obtain the following corollary.
Corollary 3.4
 (i)
g is continuous and \(g(X)\) is closed;
 (ii)
\(F(X\times X)\subset g(X)\) and g and F are compatible;
 (iii)
there exist \(x_{0},y_{0}\in X\) such that \(gx_{0}\preceq F(x_{0},y_{0})\) and \(gy_{0}\preceq F(y_{0},x_{0})\);
 (iv)there exists \(k\in[0,1)\) such that for all \(x,y,u,v\in X\) satisfying (\(gx\preceq gu\) and \(gy\preceq gv\)) or (\(gu\preceq gx\) and \(gv\preceq gy\)),holds true;$$ d\bigl(F(x,y),F(u,v)\bigr)\leq k\max\bigl\{ d(gx,gu),d(gy,gv)\bigr\} $$(3.10)
 (v)
(a) F is continuous or (b) if \(\{x_{n}\}\) is an increasing sequence in X and \(x_{n}\to x\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all n.
Corollary 3.5
 (i)
F satisfies the monotone property;
 (ii)
there exist \(x_{0},y_{0}\in X\) such that \(x_{0}\preceq F(x_{0},y_{0})\) and \(y_{0}\preceq F(y_{0},x_{0})\);
 (iii)there exists \(k\in[0,1)\) such that for all \(x,y,u,v\in X\) satisfying (\(x\preceq u\) and \(y\preceq v\)) or (\(u\preceq x\) and \(v\preceq y\)),holds true;$$ d\bigl(F(x,y),F(u,v)\bigr)\leq k\max\bigl\{ d(x,u),d(y,v)\bigr\} $$
 (iv)
(a) F is continuous or (b) if \(\{x_{n}\}\) is an increasing sequence in X and \(x_{n}\to x\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all n.
Remark 3.6
Now, reasoning on Theorem 3.1, some questions arise naturally. To be precise, one can ask the following.
Question
Is it possible to find a sufficient condition for the existence and uniqueness of the coupled common fixed point of g and F?
Motivated by the interest in this research, we give a positive answer to this question adding to Theorem 3.1 some hypotheses.
Theorem 3.7
 (vi)
for any two elements \((x,y),(u,v)\in X\times X\), there exists \((w,z)\in X\times X\) such that \((F(w,z),F(z,w))\) is comparable to both \((F(x,y),F(y,x))\) and \((F(u,v),F(v,u))\).
Proof
Theorem 3.1 implies that there exists a coupled coincidence point \((x,y)\in X\times X\), that is, \(gx=F(x,y)\) and \(gy=F(y,x)\). Suppose that there exists another coupled coincidence point \((u,v)\in X\times X\) and hence \(gu=F(u,v)\) and \(gv=F(v,u)\). We will prove that \(gx=gu\) and \(gy=gv\).
Corollary 3.8
 (vi)
for any two elements \((x,y),(u,v)\in X\times X\), there exists \((w,z)\in X\times X\) such that \((F(w,z),F(z,w))\) is comparable to both \((F(x,y),F(y,x))\) and \((F(u,v),F(v,u))\).
Theorem 3.9
In addition to the hypotheses of Corollary 3.3, let the condition (vi) of Theorem 3.7 be satisfied. Then the coupled fixed point of F is unique. Moreover, if for the terms of sequences \(\{x_{n}\}\), \(\{y_{n}\}\) defined by \(x_{n}=F(x_{n1},y_{n1})\) and \(y_{n}=F(x_{n1},y_{n1})\), \(x_{n}\preceq y_{n}\) holds for n sufficiently large, then the coupled fixed point of F has the form \((x,x)\).
Proof
Corollary 3.10
In addition to the hypotheses of Corollary 3.5, let the condition (vi) of Theorem 3.7 be satisfied. Then the coupled fixed point of F is unique. Moreover, if for the terms of sequences \(\{x_{n}\}\), \(\{y_{n}\}\) defined by \(x_{n}=F(x_{n1},y_{n1})\) and \(y_{n}=F(x_{n1},y_{n1})\), \(x_{n}\preceq y_{n}\) holds for all n, then the coupled fixed point of F has the form \((x,x)\).
Finally, we give an example showing that our theorem can be used when many results in this field cannot.
Example 3.11
Next, we show that Theorem 3.7 can be used in this example.
We can easily check that all other conditions of Theorem 3.7 hold. Therefore, g and F have a unique common coupled fixed point. In this example, we can see that a point \((0,0) \in X\times X\) is a unique common coupled fixed point of g and F.
Remark 3.12
Results concerning these notions can be found, e.g., in [24]. Such an approach is not possible for mixed monotone mappings when \(n=3\) (see [26]).
Declarations
Acknowledgements
The first author is thankful to the Ministry of Education, Science and Technological Development of Serbia, Project No. 174002. The second author was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU) and Theoretical and Computational Science Center (TaCS). The fourth 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.
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.
Authors’ Affiliations
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