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Fixed point results for ${G}^{m}$MeirKeeler contractive and $G\text{}(\alpha ,\psi )$MeirKeeler contractive mappings
Fixed Point Theory and Applications volume 2013, Article number: 34 (2013)
Abstract
In this paper, first we introduce the notion of a ${G}^{m}$MeirKeeler contractive mapping and establish some fixed point theorems for the ${G}^{m}$MeirKeeler contractive mapping in the setting of Gmetric spaces. Further, we introduce the notion of a ${G}_{c}^{m}$MeirKeeler contractive mapping in the setting of Gcone metric spaces and obtain a fixed point result. Later, we introduce the notion of a $G\text{}(\alpha ,\psi )$MeirKeeler contractive mapping and prove some fixed point theorems for this class of mappings in the setting of Gmetric spaces.
MSC:46N40, 47H10, 54H25, 46T99.
1 Introduction
In nonlinear functional analysis, the study of fixed points of given mappings satisfying certain contractive conditions in various abstract spaces has been at the center of vigorous research activity in the last decades. The Banach contraction mapping principle is one of the initial and crucial results in this direction: In a complete metric space each contraction has a unique fixed point. Following this celebrated result, many authors have devoted their attention to generalizing, extending and improving this theory. For this purpose, the authors consider to extend some wellknown results to different abstract spaces such as symmetric spaces, quasimetric spaces, fuzzy metric, partial metric spaces, probabilistic metric spaces and a Gmetric space (see, e.g., [1–9]). Several authors have reported interesting (common) fixed point results for various classes of functions in the setting of such abstract spaces (see, e.g., [6, 7, 10–32]).
In this paper, we consider especially a Gmetric space and cone metric spaces which are introduced by MustafaSims [9] and HuangZhang [3], respectively. Roughly speaking, a Gmetric assigns a real number to every triplet of an arbitrary set. On the other hand, a cone metric space is obtained by replacing the set of real numbers by an ordered Banach space. Very recently, a number of papers on these concepts have appeared [9, 33–48].
One of the remarkable notions in fixed point theory is MeirKeeler contractions [49] which have been studied by many authors (see, e.g., [50–56]). In this paper, first we introduce the notion of a ${G}^{m}$MeirKeeler contractive mapping and establish some fixed point theorems for the ${G}^{m}$MeirKeeler contractive mapping in the setting of Gmetric spaces. In Section 4, we introduce the notion of a ${G}_{c}^{m}$MeirKeeler contractive mapping in the setting of cone Gmetric spaces and establish a fixed point result. Later, we introduce the notion of a $G\text{}(\alpha ,\psi )$MeirKeeler contractive mapping and prove some fixed point theorems for this class of mappings in the setting of Gmetric spaces.
2 Preliminaries
We present now the necessary definitions and results in Gmetric spaces which will be useful; for more details, we refer to [9, 57]. In the sequel, ℝ, ${\mathbb{R}}_{+}$ and ℕ denote the set of real numbers, the set of nonnegative real numbers and the set of positive integers, respectively.
Definition 1 Let X be a nonempty set. A function $G:X\times X\times X\u27f6{\mathbb{R}}_{+}$ is called a Gmetric if the following conditions are satisfied:

(G1)
if $x=y=z$, then $G(x,y,z)=0$;

(G2)
$0<G(x,y,y)$ for any $x,y\in X$ with $x\ne y$;

(G3)
$G(x,x,y)\le G(x,y,z)$ for any points $x,y,z\in X$, with $y\ne z$;

(G4)
$G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots $ , symmetry in all three variables;

(G5)
$G(x,y,z)\le G(x,a,a)+G(a,y,z)$ for any $x,y,z,a\in X$.
Then the pair $(X,G)$ is called a Gmetric space.
Definition 2 Let $(X,G)$ be a Gmetric space, and let $\{{x}_{n}\}$ be a sequence of points of X. A point $x\in X$ is said to be the limit of the sequence $\{{x}_{n}\}$ if ${lim}_{n,m\to +\mathrm{\infty}}G(x,{x}_{m},{x}_{n})=0$, and we say that the sequence $\{{x}_{n}\}$ is Gconvergent to x and denote it by ${x}_{n}\u27f6x$.
We have the following useful results.
Proposition 3 (see [44])
Let $(X,G)$ be a Gmetric space. Then the following are equivalent:

(1)
$\{{x}_{n}\}$ is Gconvergent to x;

(2)
${lim}_{n\to +\mathrm{\infty}}G({x}_{n},{x}_{n},x)=0$;

(3)
${lim}_{n\to +\mathrm{\infty}}G({x}_{n},x,x)=0$.
Definition 4 ([44])
Let $(X,G)$ be a Gmetric space, the sequence $\{{x}_{n}\}$ is called GCauchy if for every $\epsilon >0$, there is $k\in \mathbb{N}$ such that $G({x}_{n},{x}_{m},{x}_{l})<\epsilon $ for all $n,m,l\ge k$, that is, $G({x}_{n},{x}_{m},{x}_{l})\to 0$ as $n,m,l\to +\mathrm{\infty}$.
Proposition 5 ([44])
Let $(X,G)$ be a Gmetric space. Then the following are equivalent:

(1)
the sequence $\{{x}_{n}\}$ is GCauchy;

(2)
for every $\epsilon >0$, there is $k\in \mathbb{N}$ such that $G({x}_{n},{x}_{m},{x}_{m})<\epsilon $ for all $n,m\ge k$.
Definition 6 ([44])
A Gmetric space $(X,G)$ is called Gcomplete if every GCauchy sequence in $(X,G)$ is Gconvergent in $(X,G)$.
Proposition 7 (see [44])
Let $(X,G)$ be a Gmetric space. Then, for any $x,y,z,a\in X$, it follows that

(i)
if $G(x,y,z)=0$, then $x=y=z$;

(ii)
$G(x,y,z)\le G(x,x,y)+G(x,x,z)$;

(iii)
$G(x,y,y)\le 2G(y,x,x)$;

(iv)
$G(x,y,z)\le G(x,a,z)+G(a,y,z)$;

(v)
$G(x,y,z)\le \frac{2}{3}[G(x,y,a)+G(x,a,z)+G(a,y,z)]$;

(vi)
$G(x,y,z)\le G(x,a,a)+G(y,a,a)+G(z,a,a)$.
Proposition 8 (see [44])
Let $(X,G)$ be a Gmetric space. Then the function $G(x,y,z)$ is jointly continuous in all three of its variables.
Now, we introduce the following notion of a ${G}^{m}$MeirKeeler contractive mapping.
Definition 9 Let $(X,G)$ be a Gmetric space. Suppose that $f:X\to X$ is a selfmapping satisfying the following condition:
For each $\epsilon >0$, there exists $\delta >0$ such that for all $x,y\in X$ and for all $m\in \mathbb{N}$, we have
Then f is called a ${G}^{m}$MeirKeeler contractive mapping.
Remark 10 If $f:X\to X$ is a ${G}^{m}$MeirKeeler contractive mapping on a Gmetric space X, then
holds for all $x,y\in X$ and for all $m\in \mathbb{N}$ when $G(x,{f}^{(m)}x,y)>0$. On the other hand, if $G(x,{f}^{(m)}x,y)=0$, by Proposition 7, $x={f}^{(m)}x=y$, and so $G(fx,{f}^{(m+1)}x,fy)=0$. Hence, for all $x,y\in X$ and for all $m\in \mathbb{N}$, we have
3 Fixed point result for ${G}^{m}$MeirKeeler contractive mappings
Now, we are ready to state and prove our main result.
Theorem 11 Let $(X,G)$ be a Gcomplete Gmetric space and let f be a ${G}^{m}$MeirKeeler contractive mapping on X. Then f has a unique fixed point.
Proof Define the sequence $\{{x}_{n}\}$ in X as follows:
Suppose that there exists ${n}_{0}$ such that ${x}_{{n}_{0}}={x}_{{n}_{0}+1}$. Since ${x}_{{n}_{0}}={x}_{{n}_{0}+1}=f{x}_{{n}_{0}}$, then ${x}_{{n}_{0}}$ is the fixed point of f. Hence, we assume that ${x}_{n}\ne {x}_{n+1}$ for all $n\in \mathbb{N}\cup \{0\}$, and so
By Remark 10 with $m=1$, we get
for all $n\in \mathbb{N}\cup \{0\}$. Define ${s}_{n}=G({x}_{n},{x}_{n+1},{x}_{n+1})$. Then $\{{s}_{n}\}$ is a strictly decreasing sequence in ${\mathbb{R}}_{+}$ and so it is convergent, say, to $s\in {\mathbb{R}}_{+}$. Now, we show that s must be equal to 0. Suppose, to the contrary, that $s>0$. Clearly, we have
Assume $\epsilon =s>0$. Then by hypothesis, there exists a convenient $\delta (\epsilon )>0$ such that (2.1) holds. On the other hand, by the definition of ε, there exists ${n}_{0}\in \mathbb{N}$ such that
Now, by condition (2.1) with $m=1$ and (3.4), we get
which contradicts (3.3). Hence $s=0$, that is, ${lim}_{n\to +\mathrm{\infty}}{s}_{n}=0$.
We will show that $\{{x}_{n}\}$ is a GCauchy sequence. For all $\epsilon >0$, by the hypothesis, there exists a suitable $\delta (\epsilon )>0$ such that (2.1) holds. Without loss of generality, we assume $\delta <\epsilon $. Since $s=0$, there exists $N\in \mathbb{N}$ such that
We assert that for any fixed $k\ge N$, the condition
holds. To prove it, we use the method of induction. By Remark 10 and (3.6), assertion (3.7) is satisfied for $l=1$. Suppose that (3.7) is satisfied for $l=1,2,\dots ,m$ for some $m\in \mathbb{N}$. Now, for $l=m+1$, using (3.6), we obtain
If $G({x}_{k1},{x}_{k+m},{x}_{k+m})\ge \epsilon $, then by (2.1) we get
and hence (3.7) is satisfied.
If $G({x}_{k1},{x}_{k+m},{x}_{k+m})=0$, then ${x}_{k1}={x}_{k+m}$ and hence ${x}_{k}=f{x}_{k1}=f{x}_{k+m}={x}_{k+m+1}$. This implies
and (3.7) is satisfied.
If $0<G({x}_{k1},{x}_{k+m},{x}_{k+m})<\epsilon $, by Remark 10, we obtain
Consequently, (3.7) is satisfied for $l=m+1$ and hence
Now, if $n>m\ge N$, by (3.9) and Proposition 7, we have
Hence, for all $m,n\ge N$, the following holds:
Thus $\{{x}_{n}\}$ is a GCauchy sequence. Since $(X,G)$ is Gcomplete, there exists $z\in X$ such that $\{{x}_{n}\}$ is Gconvergent to z. Now, by Remark 10 with $m=1$, we have
By taking the limit as $n\to +\mathrm{\infty}$ in the above inequality and using the continuity of the function G, we get
and hence, $z=fz$, that is, z is a fixed point of f. To prove the uniqueness, we assume that $w\in X$ is another fixed point of f such that $z\ne w$. Then $G(z,{f}^{(m)}z,w)=G(z,z,w)>0$. Now, by Remark 10, we get
which is a contradiction and hence $z=w$. □
Example 12 Let $X=[0,\mathrm{\infty})$ and
be a Gmetric on X. Define $f:X\to X$ by $fx=\frac{1}{2}x$. Then ${f}^{m}x=\frac{1}{{2}^{m}}x$. Assume that $x\le y$. Then
and
Let, $\u03f5>0$. Then, for any $\delta =\u03f5$, condition (2.1) holds. Similarly, condition (2.1) holds when $y\le x$. That is, f is a ${G}^{m}$MeirKeeler contractive mapping. The condition of Theorem 11 holds, and so f has a unique fixed point.
4 Fixed point for $G\text{}(\alpha ,\psi )$MeirKeeler contractive mappings
In this section we introduce a notion of a $G\text{}(\alpha ,\psi )$MeirKeeler contractive mapping and establish some results of a fixed point for such class of mappings.
Denote with Ψ the family of nondecreasing functions $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ continuous in $t=0$ such that

$\psi (t)=0$ if and only if $t=0$;

$\psi (t+s)\le \psi (t)+\psi (s)$.
Samet, Vetro and Vetro [19] introduced the following concept.
Definition 13 Let $f:X\to \phantom{\rule{0.25em}{0ex}}X$ and $\alpha :X\times X\to {\mathbb{R}}_{+}$. We say that f is an αadmissible mapping if
Now, we apply this concept in the following definition.
Definition 14 Let $(X,G)$ be a Gmetric space and $\psi \in \mathrm{\Psi}$. Suppose that $f:X\to X$ is an αadmissible mapping satisfying the following condition:
For each $\epsilon >0$, there exists $\delta >0$ such that
for all $x,y,z\in X$. Then f is called a $G\text{}(\alpha ,\psi )$MeirKeeler contractive mapping.
Remark 15 Let f be a $G\text{}(\alpha ,\psi )$MeirKeeler contractive mapping. Then
for all $x,y\in X$ when $G(x,y,z)>0$. Also, if $G(x,y,z)=0$, then $x=y=z$, which implies $G(fx,fy,fz)=0$, i.e.,
for all $x,y,z\in X$.
Theorem 16 Let $(X,G)$ be a Gcomplete Gmetric space. Suppose that f is a continuous $G\text{}(\alpha ,\psi )$MeirKeeler contractive mapping and that there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},{x}_{0})\ge 1$. Then f has a fixed point.
Proof Let ${x}_{0}\in X$ and define the sequence $\{{x}_{n}\}$ by ${x}_{n}={f}^{n}{x}_{0}$ for all $n\in \mathbb{N}$. Since f is an αadmissible mapping and $\alpha ({x}_{0},{x}_{0})\ge 1$, we deduce that $\alpha ({x}_{1},{x}_{1})=\alpha (f{x}_{0},f{x}_{0})\ge 1$. By continuing this process, we get $\alpha ({x}_{n},{x}_{n})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$. If ${x}_{{n}_{0}}={x}_{{n}_{0}+1}$ for some ${n}_{0}\in \mathbb{N}\cup \{0\}$, then obviously f has a fixed point. Hence, we suppose that
for all $n\in \mathbb{N}\cup \{0\}$. By (G2), we have
for all $n\in \mathbb{N}\cup \{0\}$. Now, define ${s}_{n}=\psi (G({x}_{n},{x}_{n+1},{x}_{n+1}))$. By Remark 15, we deduce that for all $n\in \mathbb{N}\cup \{0\}$,
which implies
Hence, the sequence $\{{s}_{n}\}$ is decreasing in ${\mathbb{R}}_{+}$ and so it is convergent to $s\in {\mathbb{R}}_{+}$. We will show that $s=0$. Suppose, to the contrary, that $s>0$. Hence, we have
Let $\epsilon =s>0$. Then by hypothesis, there exists a $\delta (\epsilon )>0$ such that (4.10) holds. On the other hand, by the definition of ε, there exists ${n}_{0}\in \mathbb{N}$ such that
Now, by (4.10) we have
which is a contradiction. Hence $s=0$, that is, ${lim}_{n\to +\mathrm{\infty}}{s}_{n}=0$. Now, by the continuity of ψ in $t=0$, we have
For given $\epsilon >0$, by the hypothesis, there exists a $\delta =\delta (\epsilon )>0$ such that (4.10) holds. Without loss of generality, we assume $\delta <\epsilon $. Since $s=0$, then there exists $N\in \mathbb{N}$ such that
We will prove that for any fixed $k\ge {N}_{0}$,
holds. Note that (4.6), by (4.5), holds for $l=1$. Suppose condition (4.10) is satisfied for some $m\in \mathbb{N}$. For $l=m+1$, by (G5) and (4.5), we get
If $\psi (G({x}_{k1},{x}_{k+m},{x}_{k+m}))\ge \epsilon $, then by (4.10) we get
and hence (4.6) holds.
If $\psi (G({x}_{k1},{x}_{k+m},{x}_{k+m}))<\epsilon $, by Remark 15, we get
Consequently, (4.6) holds for $l=m+1$. Hence, $\psi (G({x}_{k},{x}_{k+l},{x}_{k+l}))\le \epsilon $ for all $k\ge {N}_{0}$ and $l\ge 1$, which means
Then, for all $n>m\ge {N}_{0}$, by (4.8) and Proposition 7, we have
That is, for all $m,n\ge {N}_{0}$, the following condition holds:
Consequently, ${lim}_{m,n\to +\mathrm{\infty}}\psi (G({x}_{n},{x}_{m},{x}_{m}))=0$. By the continuity of ψ in $t=0$, we get ${lim}_{n\to +\mathrm{\infty}}G({x}_{n},{x}_{m},{x}_{m})=0$. Hence $\{{x}_{n}\}$ is a GCauchy sequence. Since $(X,G)$ is Gcomplete, there exists $z\in X$ such that
Also, by the continuity of f, we have
and hence
that is, $z=fz$. □
Theorem 17 Let $(X,G)$ be a Gcomplete Gmetric space and let f be a $G\text{}(\alpha ,\psi )$MeirKeeler contractive mapping. If the following conditions hold:

(i)
there exists ${x}_{0}\in X$ such that $\alpha ({x}_{0},{x}_{0})\ge 1$;

(ii)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n})\ge 1$ for all n and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\alpha (x,x)\ge 1$,
then f has a fixed point.
Proof Let ${x}_{0}\in X$ such that $\alpha ({x}_{0},{x}_{0})\ge 1$. Define the sequence $\{{x}_{n}\}$ in X by ${x}_{n}={f}^{n}{x}_{0}$ for all $n\in \mathbb{N}$. Following the proof of Theorem 16, we say that $\alpha ({x}_{n},{x}_{n})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and that there exists $z\in X$ such that ${x}_{n}\to z$ as $n\to +\mathrm{\infty}$. Hence, from (ii) $\alpha (z,z)\ge 1$. By Remark 15, we have
By taking limit as $n\to +\mathrm{\infty}$, in the above inequality, we get $\psi (G(fz,z,z))\le 0$, that is, $G(fz,z,z)=0$. Hence $fz=z$. □
Theorem 18 Assume that all the hypotheses of Theorem 16 (and 17) hold. Adding the following conditions:

(iii)
$\alpha (z,z)\ge 1$ for all $z\in X$,
we obtain the uniqueness of the fixed point of f.
Proof Suppose that z and ${z}^{\ast}$ are two fixed points of f such that $z\ne {z}^{\ast}$. Then $G({z}^{\ast},z,z)>0$. Now, by Remark 15, we have
which is a contradiction. Hence, $z={z}^{\ast}$. □
If in Theorems 17 and 18 we take $\alpha (x,y)=a$ and $\psi (t)=t$ where $a\ge 1$, then we have the following corollary.
Corollary 19 Let $(X,G)$ be a Gcomplete Gmetric space. Suppose that $f:X\to X$ is a mapping satisfying the following condition:
For each $\epsilon >0$, there exists $\delta >0$ such that
for all $x,y,z\in X$ where $a\ge 1$. Then f has a unique fixed point.
5 Fixed point in Gcone metric spaces
In this section we recall the notion of a cone Gmetric [36], we introduce the notion of a ${G}_{c}^{m}$MeirKeeler contractive mapping and establish the result of a fixed point for such class of mappings.
Definition 20 ([3])
Let E be a real Banach space with θ as the zero element and with the norm $\parallel \cdot \parallel $. A subset P of E is called a cone if and only if the following conditions are satisfied:

(i)
P is closed, nonempty and $P\ne \{\theta \}$;

(ii)
$a,b\ge 0$ and $x\in P$ implies $ax+by\in P$;

(iii)
$x\in P$ and $x\in P$ implies $x=\theta $.
Let $P\subset E$ be a cone, we define a partial ordering ⪯ on E with respect to P by $x\u2aafy$ if and only if $yx\in P$; we write $x\prec y$ whenever $x\u2aafy$ and $x\ne y$, while $x\ll y$ will stand for $yx\in intP$ (the interior of P). The cone $P\subset E$ is called normal if there is a positive real number K such that for all $x,y\in E$, $\theta \u2aafx\u2aafy\Rightarrow \parallel x\parallel \le K\parallel y\parallel $. The least positive number satisfying the above inequality is called the normal constant of P. If $K=1$, then the cone P is called monotone.
Definition 21 Let $(E,\parallel \cdot \parallel )$ be a real Banach space with a monotone solid cone P. A mapping ${G}_{c}:X\times X\times X\u27f6E$ satisfying the following conditions:

(F1)
if $x=y=z$, then ${G}_{c}(x,y,z)=\theta $;

(F2)
$\theta \ll {G}_{c}(x,y,y)$ for any $x,y\in X$ with $x\ne y$;

(F3)
${G}_{c}(x,x,y)\u2aaf{G}_{c}(x,y,z)$ for any points $x,y,z\in X$, with $y\ne z$;

(F4)
${G}_{c}(x,y,z)={G}_{c}(x,z,y)={G}_{c}(y,z,x)=\cdots $ , symmetry in all three variables;

(F5)
${G}_{c}(x,y,z)\u2aaf{G}_{c}(x,a,a)+{G}_{c}(a,y,z)$ for any $x,y,z,a\in X$
is a cone Gmetric on X and $(X,{G}_{c})$ is a cone Gmetric space.
Let $(E,\parallel \cdot \parallel )$ be a real Banach space with a monotone solid cone P. Then
Proposition 23 ([8])
Let $(E,\parallel \cdot \parallel )$ be a real Banach space with a monotone solid cone P. If ${G}_{c}:X\times X\times X\u27f6E$ is a Gcone metric on X, then the function $G:X\times X\times X\u27f6[0,+\mathrm{\infty})$ defined by $G(x,y,z)=\parallel {G}_{c}(x,y,z)\parallel $ is a Gmetric on X and $(X,G)$ a Gmetric space.
Definition 24 Let $(E,\parallel \cdot \parallel )$ be a real Banach space with a monotone solid cone P and $(X,{G}_{c})$ be a cone Gmetric space. Suppose that $f:X\to X$ is a selfmapping satisfying the following condition:
For each $\mathrm{\Upsilon}\in intP$, there exists $\mathrm{\Delta}\in intP$ such that for all $x,y\in X$ and for all $m\in \mathbb{N}$,
Then f is called a ${G}_{c}^{m}$MeirKeeler contractive mapping.
Theorem 25 Let $(E,\parallel \cdot \parallel )$ be a real Banach space with a monotone solid cone P and $(X,{G}_{c})$ be a Gcomplete Gcone metric space and f be a ${G}_{c}^{m}$MeirKeeler contractive mapping on X. Then f has a unique fixed point.
Proof For a given $\epsilon >0$, let $\epsilon \le G(x,{f}^{(m)}x,y)$, where $G=\parallel {G}_{c}\parallel $. This implies
for given $H\in intP$. Indeed, if $\frac{\epsilon H}{\parallel H\parallel}{G}_{c}(x,{f}^{(m)}x,y)\in intP$, then
and so by Lemma 22, we get $G(x,{f}^{(m)}x,y)<\epsilon $, which is a contradiction. Therefore (5.2) holds.
Now suppose that $G(x,{f}^{(m)}x,y)<\epsilon +\delta $. This implies
Indeed if
then
and so $\epsilon +\delta \le G(x,{f}^{(m)}x,y)$, which is a contradiction. This implies that (5.3) holds.
Now, by (5.4), (5.2) and (5.3), we have
Again, by Lemma 22, we get
Thus f is a ${G}^{m}$MeirKeeler contractive mapping, and by Theorem 11, f has a unique fixed point. □
Similarly, we have the following corollary.
Corollary 26 Let $(E,\parallel \cdot \parallel )$ be a real Banach space with a monotone solid cone P and $(X,{G}_{c})$ be a Gcomplete Gcone metric space and f be a mapping such that for each $\mathrm{\Upsilon}\in intP$, there exists $\mathrm{\Delta}\in intP$ such that
for all $x,y\in X$, where $a\ge 1$. Then f has a unique fixed point.
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Acknowledgements
This research was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR, KAU for financial support. The authors would like to thank the editor and the referees for their suggestions to improve the presentation of the paper. The 3rd author is thankful for support of Islamic Azad University, Astara, during this research.
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Hussain, N., Karapınar, E., Salimi, P. et al. Fixed point results for ${G}^{m}$MeirKeeler contractive and $G\text{}(\alpha ,\psi )$MeirKeeler contractive mappings. Fixed Point Theory Appl 2013, 34 (2013). https://doi.org/10.1186/16871812201334
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Keywords
 ${G}^{m}$MeirKeeler contractive mapping
 Gmetric space
 ${G}_{c}^{m}$MeirKeeler contractive mapping
 Gcone metric space
 $G\text{}(\alpha ,\psi )$MeirKeeler contractive mapping