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 Open Access
Fixed point results for ${G}^{m}$MeirKeeler contractive and $G\text{}(\alpha ,\psi )$MeirKeeler contractive mappings
 Nawab Hussain^{1}Email author,
 Erdal Karapınar^{2},
 Peyman Salimi^{3} and
 Pasquale Vetro^{4}
https://doi.org/10.1186/16871812201334
© Hussain et al.; licensee Springer. 2013
 Received: 15 November 2012
 Accepted: 6 February 2013
 Published: 21 February 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.
Keywords
 ${G}^{m}$MeirKeeler contractive mapping
 Gmetric space
 ${G}_{c}^{m}$MeirKeeler contractive mapping
 Gcone metric space
 $G\text{}(\alpha ,\psi )$MeirKeeler contractive mapping
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.
 (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])
 (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])
 (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])
 (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:
Then f is called a ${G}^{m}$MeirKeeler contractive mapping.
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.
which contradicts (3.3). Hence $s=0$, that is, ${lim}_{n\to +\mathrm{\infty}}{s}_{n}=0$.
and hence (3.7) is satisfied.
and (3.7) is satisfied.
which is a contradiction and hence $z=w$. □
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.
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 all $x,y,z\in X$. Then f is called a $G\text{}(\alpha ,\psi )$MeirKeeler contractive mapping.
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.
and hence (4.6) holds.
that is, $z=fz$. □
 (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.
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 $fz=z$. □
 (iii)
$\alpha (z,z)\ge 1$ for all $z\in X$,
we obtain the uniqueness of the fixed point of f.
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 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])
 (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.
 (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.
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:
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.
and so by Lemma 22, we get $G(x,{f}^{(m)}x,y)<\epsilon $, which is a contradiction. Therefore (5.2) holds.
and so $\epsilon +\delta \le G(x,{f}^{(m)}x,y)$, which is a contradiction. This implies that (5.3) holds.
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.
for all $x,y\in X$, where $a\ge 1$. Then f has a unique fixed point.
Declarations
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.
Authors’ Affiliations
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