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A shortnote on ‘Common fixed point theorems for noncompatible selfmaps in generalized metric spaces’
Journal of Inequalities and Applications volume 2015, Article number: 55 (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.
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
The selfmappings f and g are said to be compatible if
whenever \(\{x_{n}\}\) is a sequence in X such that
Definition 2
The selfmappings f and g are said to be Rweakly commuting mappings of type \((A_{g})\) if there exists some positive real number R such that
One of the main results in [28] is the following one.
Theorem 3
(Yang [28, Theorem 2.1])
Let \((X,G)\) be a Gmetric space and \((f,g)\) be a pair of noncompatible selfmappings with \(\overline{fX}\subset gX\) (here \(\overline{fX}\) denotes the closure of fX). Assume the following conditions are satisfied:
for all \(x,y,z\in X\). Here \(\alpha\in [ 0,1 ) \). If \((f,g)\) are a pair of Rweakly commuting mappings of type \((A_{g})\), then f and g have a unique common fixed point (say t) and both f and g are not Gcontinuous at t.
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
Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be two sequences of a Gmetric space \(( X,G ) \). Then
On the other hand, the author assumed in Theorem 3 that f and g are not compatible. In such a case, there exists a sequence \(\{x_{n}\}\subseteq X\), such that
but either
does or does not exist and if it does it is different from zero. Even avoiding condition (4), this property was introduced by Aamri and El Moutawakil in [29] in the context of metric spaces.
Definition 5
(Aamri and El Moutawakil [29])
Let \(f,g:X\rightarrow X\) be two selfmappings of a metric space \((X,d)\). We say that f and g satisfy the (E.A.) property if there exist a sequence \(\{x_{n}\}\subseteq X\) and a point \(t\in X\) such that
In the framework of Gmetric spaces, we have the following analog.
Definition 6
(Mustafa et al. [30])
Let \(f,g:X\rightarrow X\) be two selfmappings of a Gmetric space \((X,G)\). We say that f and g satisfy the (E.A.) property if there exist a sequence \(\{x_{n}\}\subseteq X\) and a point \(t\in X\) such that
Also in Theorem 3, the author assumed that \(\overline {fX}\subset gX\). In such a case, the limit verifies
As a consequence, there exists \(u\in X\) such that \(t=gu\). This idea yields the following notion, called common limit in the range of g, which originally was introduced by Sintunaravat and Kumam in [31] in the context of fuzzy metric spaces and, later, was particularized to Gmetric spaces by Aydi et al. in [32].
Definition 7
(Aydi et al. [32])
Let \(f,g:X\rightarrow X\) be two selfmappings of a Gmetric space \((X,G)\). We say that f and g satisfy the ‘common limit in the range of g’ property (briefly, \((CLRg)\)property) if there exist a sequence \(\{x_{n}\}\subseteq X\) and a point \(u\in X\) such that
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.
Before that, we must point out that the author did not appropriately take limit in the inequalities throughout the paper. Let us show some examples. Following the lines of Theorem 2.1 in [28], as \(t\in\overline {fX}\subset gX\), there exists \(u\in X\) such that \(gu=t\). Applying the contractivity condition (3) to \(x=u\) and \(y=z=x_{n}\), the author wrote (see [28, p.4, lines 1819]):
Letting \(n\rightarrow\infty\), the author wrote (see [28, p.4, lines 2122]):
Unfortunately, inequality (7) is false, because the author seems to apply that \(\{x_{n}\}\rightarrow u\) and f and g are continuous. This is not the case, because we only know that
In such a case, letting \(n\rightarrow\infty\) in (6), we obtain
This correct inequality is better because we may assume that \(\alpha\in [ 0,2 ) \) to deduce that \(fu=gu\). In other words, as the reader can easily see, we can refine the arguments in [28] to get sharper results. This is the main aim of the present manuscript.
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
Let \((X,G)\) be a Gmetric space and let \(f,g:X\rightarrow X\) be two selfmappings satisfying the \((CLRg)\)property. Suppose that there exists \(\alpha\in [ 0,1 ) \) such that
for all \(x,y,z\in X\). Then any point \(u\in X\) as in (5) is a coincidence point of f and g, that is, \(fu=gu\).
Proof
As \(( f,g ) \) satisfies the \((CLRg)\)property, there exist a sequence \(\{x_{n}\}\subseteq X\) and a point \(u\in X\) such that
Let us apply the contractivity condition using \(x=u\) and \(y=x_{n}\). Then, for all \(n\in\mathbb{N}\), it follows that
Taking into account (9) and the fact that G is jointly continuous on its three variables, then, letting \(n\rightarrow\infty\) in (10), we deduce that
As \(\alpha\in [ 0,1 ) \), then \(G(fu,gu,gu)=0\), so \(fu=gu\). □
If the contractivity condition is slightly stronger, then it is easy to show a second part.
Theorem 10
Let \((X,G)\) be a Gmetric space and let \(f,g:X\rightarrow X\) be two selfmappings satisfying the \((CLRg)\)property. Suppose that there exists \(\alpha\in [ 0,1 ) \) such that
for all \(x,y\in X\). Then any point \(u\in X\) as in (5) is a coincidence point of f and g, that is, \(fu=gu\).
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 \(\omega=fu=gu\).
And if we additionally assume that f is Gcontinuous at ω, then
whatever the sequence \(\{x_{n}\}\) as in (5).
Proof
Taking into account that
then condition (11) implies condition (8). As a consequence, Theorem 9 guarantees that any point \(u\in X\) as in (5) is a coincidence point of f and g, that is,
Next, assume that \((f,g)\) is a pair of Rweakly commuting mappings of type \((A_{g})\). In such a case,
Therefore,
Let us apply the contractivity condition (11) to \(x=u\) and \(y=fu\). Then we deduce
By (12) and (13), it follows that
which means that
If we take \(\omega=fu=gu\), then
so ω is a common fixed point of f and g.
Next we show that the common fixed point ω is unique. Actually, suppose that \(z\in X\) is also a common fixed point of f and g. Then, by the contractivity condition (11) applied to \(x=\omega\) and \(y=z\), we derive that
which means that \(\omega=z\).
Finally, assume that f is Gcontinuous at ω. Therefore, as \(\{fx_{n}\}\rightarrow gu=\omega\) and \(\{gx_{n}\}\rightarrow gu=\omega \),
Moreover, as \((f,g)\) is a pair of Rweakly commuting mappings of type \((A_{g})\),
Hence, for all \(n\in\mathbb{N}\), we have
Taking the limit as \(k\rightarrow\infty\) we deduce that
and, by Proposition 4, we conclude that
which means that f and g are compatible. □
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.
Proof
It follows from the fact that (3) implies (11) using \(y=z\). □
In the sequel, we extend the previous results. Let
It is clear that, given \(\alpha\in [ 0,1 ) \), the mapping \(\phi_{\alpha}:[0,\infty)\rightarrow{}[0,\infty)\) defined by \(\phi _{\alpha}(s)=\alpha s\) for all \(s\in{}[0,\infty)\), belongs to ℱ.
Theorem 13
Let \((X,G)\) be a Gmetric space and let \(f,g:X\rightarrow X\) be two selfmappings satisfying the \((CLRg)\)property. Suppose that there exists \(\phi\in\mathcal{F}\) such that
for all \(x,y\in X\). Then any point \(u\in X\) as in (5) is a coincidence point of f and g, that is, \(fu=gu\).
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 \(\omega=fu=gu\).
And if we additionally assume that f is Gcontinuous at ω, then
whatever the sequence \(\{x_{n}\}\) as in (5), that is, f and g are compatible.
Proof
For convenience, let us define, for all \(x,y\in X\),
Hence, the contractivity condition (14) can be rewritten as
As \(( f,g ) \) satisfies the \((CLR{g})\)property, there exist a sequence \(\{x_{n}\}\subseteq X\) and a point \(u\in X\) such that
We prove that \(fu=gu\) by reductio ad absurdum, that is, we assume that \(fu\neq gu\) and we will get a contradiction. In such a case,
Let us apply the contractivity condition (14) using \(x=u\) and \(y=x_{n}\). Then, for all \(n\in\mathbb{N}\), it follows that
where
We can distinguish two cases.

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

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 have
$$\lim_{n\rightarrow\infty}M ( u,x_{n} ) =G(fu,gu,gu) \quad\mbox{and} \quad M ( u,x_{n} ) >G(fu,gu,gu)\quad\mbox{for all }n\geq n_{0}. $$Hence, as \(\phi\in\mathcal{F}\), it follows from (16) that
$$\begin{aligned} G(fu,gu,gu)&=\lim_{n\rightarrow\infty}G(fu,fx_{n},fx_{n}) \leq\lim_{n\rightarrow\infty}\phi \bigl( M ( u,x_{n} ) \bigr) \\ &=\lim _{s\rightarrow G(fu,gu,gu)^{+}}\phi ( s ) < G(fu,gu,gu), \end{aligned}$$which is also a contradiction.
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.
Next, assume that \((f,g)\) is a pair of Rweakly commuting mappings of type \((A_{g})\). In such a case,
Therefore,
Let us apply the contractivity condition (14) to \(x=u\) and \(y=fu\). Then we deduce
where
As a consequence,
If \(fu\neq ffu\), then
which is impossible. Then, necessarily,
If we take \(\omega=fu=gu\), then
so ω is a common fixed point of f and g.
Next we show that the common fixed point ω is unique. Actually, suppose that \(z\in X\) is also a common fixed point of f and g. Then, by the contractivity condition (14) applied to \(x=\omega\) and \(y=z\), we derive that
where
The condition
implies that \(G(\omega,z,z)=0\), which means that \(\omega=z\).
Finally, assume that f is Gcontinuous at ω. Therefore, as \(\{fx_{n}\}\rightarrow gu=\omega\) and \(\{gx_{n}\}\rightarrow gu=\omega \),
Moreover, as \((f,g)\) is a pair of Rweakly commuting mappings of type \((A_{g})\),
Hence, for all \(n\in\mathbb{N}\), we have
Taking the limit as \(k\rightarrow\infty\) we deduce that
and, by Proposition 4, we conclude that
which means that f and g are compatible. □
Taking into account that
then Theorem 3 (avoiding the unproved fact that g is not Gcontinuous at the unique common fixed point) is an immediate consequence of Theorem 13.
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
Let \(X= [ 0,\infty ) \) be endowed with the complete Gmetric \(G(x,y,z)=\vert xy\vert +\vert xz\vert +\vert yz\vert \) for all \(x,y,z\in X\), and let us consider the mappings \(f,g:X\rightarrow X\) defined by \(fx=x\) and \(gx=2x\) for all \(x\in X\). The sequence \(x_{n}=1/n\) for all \(n\geq1\) shows that f and g satisfy the \((CLRg)\)property. Furthermore, for all \(x,y\in X\), we have
Then Theorem 10 guarantees that f and g have a coincidence point (and so does Theorem 13). In fact, as f is the identity mapping on X, trivially f and g are Rweakly commuting mappings of type \((A_{g})\) for \(R=1\), so f and g have a unique common fixed point, which is \(\omega=0\). In addition to this, as f is continuous, f and g are compatible. Nevertheless, as f and g are compatible and continuous, Theorem 3 is not applicable.
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
Let \(X= [ 0,5 ] \) be endowed with the complete Gmetric \(G(x,y,z)=\vert xy\vert +\vert xz\vert +\vert yz\vert \) for all \(x,y,z\in X\), and let us consider the mappings \(f,g:X\rightarrow X\) defined, for all \(x\in X\), by
The sequence \(x_{n}=1+1/n\) for all \(n\geq1\) shows that f and g satisfy the \((CLRg)\)property because
We claim that the contractivity condition (11) is satisfied using \(\alpha=1/2\). Indeed, on the one hand, we have, for all \(x,y\in X\),
We only have to discuss the cases in which \(G(fx,fy,fy)\) takes the value 2. We distinguish the following possibilities.

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
$$2G(fy,gy,gy)=2 G(1,2,2)=4=\frac{1}{2} 2=\alpha G(fx,fy,fy). $$
In any case, the contractivity condition (11) holds. Furthermore, as
f and g are Rweakly commuting mappings of type \((A_{g})\), where \(R=1\). As a consequence, Theorem 10 guarantees that f and g have a unique common fixed point (and so does Theorem 13).
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Acknowledgements
AR has been partially supported by Junta de Andalucía by project FQM268 of the Andalusian CICYE.
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RoldánLópezdeHierro, A., Karapınar, E. & Alsulami, H.H. A shortnote on ‘Common fixed point theorems for noncompatible selfmaps in generalized metric spaces’. J Inequal Appl 2015, 55 (2015). https://doi.org/10.1186/s1366001505796
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MSC
 46T99
 47H10
 47H09
 54H25
Keywords
 common fixed point
 contractivity condition
 compatible mappings
 weakly commuting mappings
 (E.A.) property
 \((CLRg)\)property