 Research
 Open Access
A shortnote on ‘Common fixed point theorems for noncompatible selfmaps in generalized metric spaces’
 AntonioFrancisco RoldánLópezdeHierro^{1}Email author,
 Erdal Karapınar^{2, 3} and
 Hamed H Alsulami^{3}
https://doi.org/10.1186/s1366001505796
© RoldánLópezdeHierro et al.; licensee Springer. 2015
 Received: 27 August 2014
 Accepted: 27 January 2015
 Published: 17 February 2015
Abstract
The main aim of this shortnote is point out that certain hypotheses assumed on some results in the very recent paper (Yang in J. Inequal. Appl. 2014:275, 2014) are unnecessary, and the results contained in that manuscript can easily be improved.
Keywords
 common fixed point
 contractivity condition
 compatible mappings
 weakly commuting mappings
 (E.A.) property
 \((CLRg)\)property
MSC
 46T99
 47H10
 47H09
 54H25
1 Introduction and preliminaries
In recent times, due to its possible application to almost all branches of numerical sciences, the researchers’ interest about fixed point theory has raised very much. Especially significant have been the fixed point results in partially ordered metric spaces (see [1, 2]), in Gmetric spaces (see [3–6]), among other abstract metric spaces (see [7, 8] in partial metric spaces, [9–11] in fuzzy metric spaces, [12, 13] in intuitionistic fuzzy metric spaces, [14, 15] in probabilistic metric spaces and [16, 17] in Menger spaces), even in the multidimensional case (see [18–25]). In this paper we focus in the setting of Gmetric spaces. Some basic notions and results about Gmetric spaces (metric structure, convergence, completeness, etc.) can be found, for instance, in [5, 6, 26, 27].
In the sequel, let \(( X,G ) \) be a Gmetric space and let \(f,g:X\rightarrow X\) be two selfmappings. In [28], the author introduced the following notions and basic facts.
Definition 1
Definition 2
One of the main results in [28] is the following one.
Theorem 3
(Yang [28, Theorem 2.1])
2 Main remarks
First of all, about the definition given by the author of compatible mappings, we must clarify that conditions (1) and (2) are equivalent. In fact, in any Gmetric space \((X,G ) \), one of the most useful properties is the well known inequality \(G(x,x,y)\leq2 G ( x,y,y ) \) for all \(x,y\in X\). As a result, the following statement is trivial.
Proposition 4
Definition 5
(Aamri and El Moutawakil [29])
In the framework of Gmetric spaces, we have the following analog.
Definition 6
(Mustafa et al. [30])
Definition 7
(Aydi et al. [32])
This conclusion also holds when gX is closed. Then we have the following properties.
Lemma 8
 (1)
\(( f,g ) \) is not compatible ⇒ \(( f,g ) \) satisfies the (E.A.)property.
 (2)
\(( f,g ) \) satisfies the (E.A.)property and gX is closed ⇒ \(( f,g ) \) satisfies the \((CLRg)\)property.
 (3)
\(( f,g ) \) satisfies the (E.A.)property and \(\overline {fX}\subset gX\) ⇒ \(( f,g ) \) satisfies the \((CLRg)\)property.
The \((CLRg)\)property has two main advantages: (1) usually, it is not necessary to assume the completeness of the Gmetric space; and (2) usually, the common limit gu is a point of coincidence of f and g, that is, \(fu=gu\). We show it in the next section.
3 Common fixed point theorems
In the following result, we improve Theorem 3 in two senses: (1) our contractivity condition is weaker; and (2) we do not assume that f and g are not compatible.
Theorem 9
Proof
If the contractivity condition is slightly stronger, then it is easy to show a second part.
Theorem 10
Furthermore, if \((f,g)\) is a pair of Rweakly commuting mappings of type \((A_{g})\), then f and g have a unique common fixed point, which is \(\omega=fu=gu\).
Proof
Remark 11
In Theorem 3, the author assumed that f and g are not compatible, and it is announced that f and g are not Gcontinuous at ω. By the previous theorem, if f and g are not compatible, then f cannot be Gcontinuous at ω. However, the argument given by the author to prove that g is not Gcontinuous at ω is not correct: assuming that g is continuous at ω, it is proved that \(\{ ffx_{n}\}\) converges to \(\omega=f\omega\), but this does not mean that f is Gcontinuous at ω (this property must be demonstrated for all sequence \(\{y_{n}\}\) converging to ω).
Corollary 12
Theorem 3 (avoiding the unproved fact that g is not Gcontinuous at the unique common fixed point) is an immediate consequence of Theorem 10.
Theorem 13
Furthermore, if \((f,g)\) is a pair of Rweakly commuting mappings of type \((A_{g})\), then f and g have a unique common fixed point, which is \(\omega=fu=gu\).
Proof

Case 1. Assume that there exists a subsequence \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(M ( u,x_{n(k)} ) \leq G(fu,gu,gu)\) for all \(k\in\mathbb{N}\). In such a case, aswe deduce that \(M ( u,x_{n(k)} ) =G(fu,gu,gu)\) for all \(k\in\mathbb{N}\). Using (16), we have$$\begin{aligned} G(fu,gu,gu) &\leq G(fu,gu,gu)+G(fx_{n(k)},gx_{n(k)},gx_{n(k)})\\ &\leq M ( u,x_{n(k)} ) \leq G(fu,gu,gu), \end{aligned}$$Taking the limit as \(k\rightarrow\infty\), we deduce that$$G(fu,fx_{n(k)},fx_{n(k)})\leq\phi \bigl( M ( u,x_{n(k)} ) \bigr) =\phi \bigl( G(fu,gu,gu) \bigr) . $$Since \(\phi\in\mathcal{F}\) and \(G(fu,gu,gu)>0\), it follows that$$G(fu,gu,gu)=\lim_{n\rightarrow\infty}G(fu,fx_{n},fx_{n}) \leq\phi \bigl( G(fu,gu,gu) \bigr) . $$which is a contradiction.$$G(fu,gu,gu)\leq\phi \bigl( G(fu,gu,gu) \bigr) < G(fu,gu,gu), $$

Case 2. Assume that there exists \(n_{0}\in\mathbb{N}\) such that \(M ( u,x_{n} ) >G(fu,gu,gu)\) for all \(n\geq n_{0}\). In such a case, we haveHence, as \(\phi\in\mathcal{F}\), it follows from (16) that$$\lim_{n\rightarrow\infty}M ( u,x_{n} ) =G(fu,gu,gu) \quad\mbox{and} \quad M ( u,x_{n} ) >G(fu,gu,gu)\quad\mbox{for all }n\geq n_{0}. $$which is also a contradiction.$$\begin{aligned} G(fu,gu,gu)&=\lim_{n\rightarrow\infty}G(fu,fx_{n},fx_{n}) \leq\lim_{n\rightarrow\infty}\phi \bigl( M ( u,x_{n} ) \bigr) \\ &=\lim _{s\rightarrow G(fu,gu,gu)^{+}}\phi ( s ) < G(fu,gu,gu), \end{aligned}$$
In any case, we get a contradiction, so we must admit that \(fu=gu\), that is, u is a coincidence point of f and g.
One of the conclusions of Theorem 3 is that f and g are not continuous at ω. Such a result is not applicable when f and g are continuous mappings, which is a very common hypothesis in fixed point theory, as in the following example.
Example 14
In the following example we illustrate the applicability of Theorems 10 and 13, and we also show that the contractivity conditions (11) and (14) are easier to prove than (3) because they only involve two variables (\(\{x,y\}\) rather than \(\{x,y,z\}\)).
Example 15

If \(0< x\leq1 \) and \(y=0\), then$$G(gx,gy,gy)=G(2,0,0)=4=\frac{1}{2} 2=\alpha G(fx,fy,fy). $$

If \(0< x\leq1 \) and \(y\in ( 1,5 ] \), then$$\begin{aligned} G(gx,fy,gy) & =G \biggl( 2,0,\frac{y1}{2} \biggr) =2+\biggl\vert 2 \frac {y1}{2}\biggr\vert +\biggl\vert \frac{y1}{2}\biggr\vert \\ & =2+2\frac{y1}{2}+\frac{y1}{2}=4=\frac{1}{2} 2=\alpha G(fx,fy,fy). \end{aligned}$$

If \(0< y\leq1 \) and \(x\in\{0\}\cup ( 1,5 ] \), then$$2G(fy,gy,gy)=2 G(1,2,2)=4=\frac{1}{2} 2=\alpha G(fx,fy,fy). $$
Declarations
Acknowledgements
AR has been partially supported by Junta de Andalucía by project FQM268 of the Andalusian CICYE.
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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