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A note on ‘Coupled fixed point theorems for mixed g-monotone mappings in partially ordered metric spaces’
- Nurcan Bilgili^{1},
- Inci M Erhan^{2}Email author,
- Erdal Karapınar^{2} and
- Duran Turkoglu^{1}
https://doi.org/10.1186/1687-1812-2014-120
© Bilgili et al.; licensee Springer. 2014
- Received: 2 January 2014
- Accepted: 29 April 2014
- Published: 15 May 2014
Abstract
Recently, some (common) coupled fixed theorems in various abstract spaces have appeared as a generalization of existing (usual) fixed point results. Unexpectedly, we noticed that most of such (common) coupled fixed theorems are either weaker or equivalent to existing fixed point results in the literature. In particular, we prove that the very recent paper of Turkoglu and Sangurlu ‘Coupled fixed point theorems for mixed g-monotone mappings in partially ordered metric spaces [Fixed Point Theory and Applications 2013, 2013:348]’ can be considered as a consequence of the existing fixed point theorems on the topic in the literature. Furthermore, we give an example to illustrate that the main results of Turkoglu and Sangurlu (Fixed Point Theory Appl. 2013:348, 2013) has limited applicability compared to the mentioned existing fixed point result.
MSC:47H10, 54H25.
Keywords
- fixed point
- G-metric space
- cyclic maps
- cyclic contractions
1 Introduction and preliminaries
In many recent publications in fixed point theory auxiliary functions are used to generalize the contractive conditions on the maps defined on various spaces. On the other hand, as appears in some studies, not all of these generalizations are meaningful. Moreover, some of the results are equivalent to, or even weaker than the existing theorems.
In this paper we discuss the insufficiency of one of these recent generalizations given by Turkoglu and Sangurlu in [1]. Our discussion can also be applied to revise some other existing results.
We first recall the definition of the following auxiliary functions and some of their basic properties.
- (1)
φ is continuous and nondecreasing,
- (2)
$\phi (t)=0$ if and only if $t=0$,
- (3)
$\phi (t+s)\le \phi (t)+\phi (s)$, $\mathrm{\forall}t,s\in [0,\mathrm{\infty})$,
Let Ψ denote all functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ which satisfy ${lim}_{t\to r}\psi (t)>0$ for all $r>0$ and ${lim}_{t\to {0}^{+}}\psi (t)=0$.
For consistency, we use the following definitions of coupled fixed point, coupled common fixed point and coupled coincidence point.
- (1)
A point $(x,y)\in X\times X$ is called a coupled fixed point of F if $x=F(x,y)$ and $y=F(y,x)$.
- (2)
A point $(x,y)\in X\times X$ is called a coupled coincidence point of F and g if $gx=F(x,y)$ and $gy=F(y,x)$.
- (3)
A point $(x,y)\in X\times X$ is called a coupled common fixed point of F if $x=gx=F(x,y)$ and $y=gy=F(y,x)$.
We next recollect the main results of Turkoglu and Sangurlu [1] by removing the typos and emphasizing the definitions of auxiliary functions which are missing in the original paper. Notice also that in the original paper of Turkoglu and Sangurlu [1], Theorem 4 is superfluous, since it is a consequence of Theorem 5.
Theorem 1.1 [1]
for all $x,y,u,v\in X$ with $gx\le gu$ and $gy\ge gv$, $F(X\times X)\subseteq g(X)$, $g(X)$ is complete and g is continuous.
- (1)
F is continuous or
- (2)
X has the following property:
- (a)
if a nondecreasing sequence $\{{x}_{n}\}\to x$, then ${x}_{n}\le x$ for all $n\in \mathbb{N}$,
- (b)
if a nonincreasing sequence $\{{y}_{n}\}\to y$, then $y\le {y}_{n}$ for all $n\in \mathbb{N}$.
Then there exist $x,y\in X$ such that $x=gx=F(x,y)$ and $y=gy=F(y,x)$, that is, F and g have a coupled common fixed point in $X\times X$.
2 Main results
In the following example, we shall emphasize the insufficiency of the main results of Turkoglu and Sangurlu [1].
Since the functions in the class Ψ are nonnegative, it is impossible to satisfy the inequality (2) for any function $\psi \in \mathrm{\Psi}$. Hence, Theorem 1.1 cannot provide the existence of a coupled common fixed point of F and g. On the other hand, it is easy to see that $(0,0)$ is a coupled common fixed point of F and g.
The weakness of Theorem 1.1 can also be observed with the following theorem.
for all $x,y,u,v\in X$ with $gx\le gu$ and $gy\ge gv$, where $F(X\times X)\subseteq g(X)$, $g(X)$ is complete and g is continuous.
- (1)
F is continuous or
- (2)
X has the following property:
- (a)
if a nondecreasing sequence $\{{x}_{n}\}\to x$, then ${x}_{n}\le x$ for all $n\in \mathbb{N}$,
- (b)
if a nonincreasing sequence $\{{y}_{n}\}\to y$, then $y\le {y}_{n}$ for all $n\in \mathbb{N}$.
Then there exist $x,y\in X$ such that $x=gx=F(x,y)$ and $y=gy=F(y,x)$, that is, F and g have a coupled common fixed point in $X\times X$.
Proof The proof of this theorem is standard. Indeed, the desired result is obtained by mimicking the lines in the proof of Turkoglu and Sangurlu [1]. Since there is no difficulty in this process, we omit the details. □
Notice that if take $g(x)=x$ in Theorem 2.1, then we derive the following result, which was proved in [2].
- (1)
F is continuous or
- (2)
X has the following property:
- (a)
if a nondecreasing sequence $\{{x}_{n}\}\to x$, then ${x}_{n}\le x$ for all $n\in \mathbb{N}$,
- (b)
if a nonincreasing sequence $\{{y}_{n}\}\to y$, then $y\le {y}_{n}$ for all $n\in \mathbb{N}$.
Then there exist $x,y\in X$ such that $x=F(x,y)$ and $y=F(y,x)$, that is, F has a coupled fixed point in $X\times X$.
Lemma 2.1 [3]
Let X be a nonempty set and $T:X\to X$ be a function. Then there exists a subset $E\subseteq X$ such that $T(E)=T(X)$ and $T:E\to X$ is one-to-one.
Theorem 2.3 Theorem 2.1 is a consequence of Theorem 2.2.
for all $gx,gy\in g(E)$. Since $g(E)=g(X)$ is complete, by using Theorem 2.2, there exist ${x}_{0},{y}_{0}\in X$ such that $G(g{x}_{0},g{y}_{0})=g{x}_{0}$ and $G(g{y}_{0},g{x}_{0})=g{y}_{0}$. Hence, F and g have a coupled coincidence point. □
for all $(x,y),(u,v)\in Y$ is a metric on Y.
The following properties can easily be seen.
- (1)
$(X,d)$ is complete if and only if $(Y,\eta )$ is complete;
- (2)
F has the mixed monotone property if and only if ${T}_{F}$ is monotone nondecreasing with respect to ⪯_{2};
- (3)
$(x,y)\in X\times X$ is a coupled fixed point of F if and only if $(x,y)$ is a fixed point of ${T}_{F}$.
- (1)
T is continuous or
- (2)
X has the following property: If a nondecreasing sequence $\{{x}_{n}\}\to x$, then ${x}_{n}\le x$ for all $n\in \mathbb{N}$.
Then there exists $x\in X$ such that $x=Tx$.
We skip the proof of Theorem 2.4 since it is standard and can be found easily in the literature; see e.g. [5]. More precisely, it is the analog of the proof given in [1].
Theorem 2.5 Theorem 2.2 follows from Theorem 2.4.
□
Theorem 2.6 Theorem 2.1 is a consequence of Theorem 2.4.
Proof Since Theorem 2.1 is a consequence of Theorem 2.2, which follows from Theorem 2.4, then Theorem 2.1 is a consequence of Theorem 2.4. □
3 Conclusion
We have presented evidence, that is, Example 2.1 and Theorem 2.1, that the results of Theorem 1.1 in [1] have more limited area of application than some existing results in the literature. Moreover, the generalizations and equivalences given in this paper can be used to show that other published theorems in the literature are in fact consequences of these generalizations. In particular, Theorem 2.11 in [6] is a consequence of Theorem 2.4. As a matter of fact, this note can be seen as a continuation of the discussion given in [4].
Declarations
Acknowledgements
The authors are thankful to the referees for careful reading of the manuscript and the valuable comments and suggestions for the improvement of the paper.
Authors’ Affiliations
References
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