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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.
References
Aydi H, Karapınar E, Salimi P: Some fixed point results in GP metric spaces. J. Appl. Math. 2012., 2012: Article ID 891713
Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, London; 2001.
Huang LG, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 2007, 332: 1468–1476. 10.1016/j.jmaa.2005.03.087
Hussain N, Abbas M: Common fixed point results for two new classes of hybrid pairs in symmetric spaces. Appl. Math. Comput. 2011, 218: 542–547. 10.1016/j.amc.2011.05.098
Hussain N, Khamsi MA, Latif A: Common fixed points for JH operators and occasionally weakly biased pairs under relaxed conditions. Nonlinear Anal. 2011, 74: 2133–2140. 10.1016/j.na.2010.11.019
Karapınar E, Salimi P: Fixed point theorems via auxiliary functions. J. Appl. Math. 2012., 2012: Article ID 792174
Matthews SG: Partial metric topology. Ann. New York Acad. Sci. 728. Proc. 8th Summer Conference on General Topology and Applications 1994, 183–197.
Moradlou, F, Salimi, P, Vetro, P: Some new extensions of EdelsteinSuzukitype fixed point theorem to Gmetric and Gcone metric spaces. Acta Math. Sci. (accepted)
Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7(2):289–297.
Aydi H, Abbas M, Vetro C: Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces. Topol. Appl. 2012, 159: 3234–3242. 10.1016/j.topol.2012.06.012
Berinde V, Vetro F: Common fixed points of mappings satisfying implicit contractive conditions. Fixed Point Theory Appl. 2012., 2012: Article ID 105
Ćirić LB: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 1974, 45: 267–273.
Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 2011, 217: 5784–5789. 10.1016/j.amc.2010.12.060
Di Bari C, Vetro P: Fixed points for weak φ contractions on partial metric spaces. Int. J. Eng., Contemp. Math. Sci. 2011, 1: 5–13.
Di Bari C, Kadelburg Z, Nashine H, Radenović S: Common fixed points of g quasicontractions and related mappings in 0complete partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 113
Paesano D, Vetro P: Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topol. Appl. 2012, 159: 911–920. 10.1016/j.topol.2011.12.008
Radenović S, Salimi P, Pantelic S, Vujaković J: A note on some tripled coincidence point results in G metric spaces. Int. J. Math. Sci. Eng. Appl. 2012, 6: 23–38. ISNN 0973–9424
Ran ACM, Reurings MC: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002993903072204
Samet B, Vetro C, Vetro P: Fixed point theorems for α  ψ contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014
Suzuki T: A new type of fixed point theorem in metric spaces. Nonlinear Anal. 2009, 71: 5313–5317. 10.1016/j.na.2009.04.017
Vetro F, Radenović S: Nonlinear ψ quasicontractions of Ćirićtype in partial metric spaces. Appl. Math. Comput. 2012. doi:10.1016/j.amc.2012.07.061
Abbas M, Khan AR, Nazir T: Coupled common fixed point results in two generalized metric spaces. Appl. Math. Comput. 2011, 217: 6328–6336. 10.1016/j.amc.2011.01.006
Abbas M, Khan AR, Nazir T: Common fixed point of multivalued mappings in ordered generalized metric spaces. Filomat 2012, 26: 1045–1053.
Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040
Sintunavarat W, Kumam P: Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces. J. Appl. Math. 2011., 2011: Article ID 637958
Abbas M, Sintunavarat W, Kumam P: Coupled fixed point of generalized contractive mappings on partially ordered G metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 31
Sintunavarat W, Kumam P: Common fixed points for R weakly commuting in fuzzy metric spaces. Ann. Univ. Ferrara 2012, 58: 389–406. 10.1007/s115650120150z
Sintunavarat W, Kumam P: Generalized common fixed point theorems in complex valued metric spaces and applications. J. Inequal. Appl. 2012., 2012: Article ID 84
Aydi H, Vetro C, Sintunavarat W, Kumam P: Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 124
Aydi H, Abbas M, Sintunavarat W, Kumam P: Tripled fixed point of W compatible mappings in abstract metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 134
Agarwal RP, Karapinar E: Remarks on some coupled fixed point theorems in G metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 2
Abbas M, Ali B, Sintunavarat W, Kumam P: Tripled fixed point and tripled coincidence point theorems in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 187
Abbas M, Nazir T, Vetro P: Common fixed point results for three maps in G metric spaces. Filomat 2011, 25(4):1–17. 10.2298/FIL1104001A
Altun I, Durmaz G: Some fixed point theorems on ordered cone metric spaces. Rend. Circ. Mat. Palermo 2009, 58: 319–325. 10.1007/s122150090026y
Aydi H, Shatanawi W, Vetro C: On generalized weak G contraction mapping in G metric spaces. Comput. Math. Appl. 2011, 62: 4223–4229.
Beg I, Abbas M, Nazir T: Generalized cone metric spaces. J. Nonlinear Sci. Appl. 2010, 3: 21–31.
Di Bari C, Vetro P: φ pairs and common fixed points in cone metric spaces. Rend. Circ. Mat. Palermo 2008, 57: 279–285. 10.1007/s1221500800209
Di Bari C, Vetro P: Weakly φ pairs and common fixed points in cone metric spaces. Rend. Circ. Mat. Palermo 2009, 58: 125–132. 10.1007/s1221500900124
Di Bari C, Vetro P: Common fixed points in cone metric spaces for MK pairs and L pairs. Ars Comb. 2011, 99: 429–437.
Di Bari C, Saadati R, Vetro P:Common fixed points in cone metric spaces for CJMpairs. Math. Comput. Model. 2011, 54: 2348–2354. 10.1016/j.mcm.2011.05.043
Farajzadeh AP, AminiHarandi A, Baleanu D: Fixed point theory for generalized contractions in cone metric space. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 708–712. 10.1016/j.cnsns.2011.01.016
Mustafa Z, Obiedat H: A fixed point theorem of Reich in G metric spaces. CUBO 2010, 12(1):83–93. 10.4067/S071906462010000100008
Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed point results in G metric spaces. Int. J. Math. Math. Sci. 2009., 2009: Article ID 283028
Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175
Saadati R, Vaezpour SM, Vetro P, Rhoades BE: Fixed point theorems in generalized partially ordered G metric spaces. Math. Comput. Model. 2010, 52: 797–801. 10.1016/j.mcm.2010.05.009
Shatanawi W: Fixed point theory for contractive mappings satisfying Φmaps in G metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 181650
Shatanawi W: Some fixed point theorems in ordered G metric spaces and applications. Abstr. Appl. Anal. 2011., 2011: Article ID 126205
Vetro P: Common fixed points in cone metric spaces. Rend. Circ. Mat. Palermo 2007, 56: 464–468. 10.1007/BF03032097
Meir A, Keeler E: A theorem on contraction mappings. J. Math. Anal. Appl. 1969, 28: 326–329. 10.1016/0022247X(69)900316
Aydi H, Vetro C, Karapınar E: MeirKeeler type contractions for tripled fixed points. Acta Math. Sci. 2012, 32(6):2119–2130.
Aydi H, Karapınar E: A MeirKeeler common type fixed point theorem on partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 26
Aydi H, Karapınar E: New MeirKeeler type tripled fixed point theorems on ordered partial metric spaces. Math. Probl. Eng. 2012., 2012: Article ID 409872
Erhan IM, Karapınar E, Türkoglu AD: Different types MeirKeeler contractions on partial metric spaces. J. Comput. Anal. Appl. 2012, 14(6):1000–1005.
Jachymski J: Equivalent conditions and the MeirKeeler type theorems. J. Math. Anal. Appl. 1995, 194(1):293–303. 10.1006/jmaa.1995.1299
Kadelburg Z, Radenović S: MeirKeelertype conditions in abstract metric spaces. Appl. Math. Lett. 2011, 24(8):1411–1414. 10.1016/j.aml.2011.03.021
Suzuki T: Fixedpoint theorem for asymptotic contractions of MeirKeeler type in complete metric spaces. Nonlinear Anal. 2006, 64: 971–978. 10.1016/j.na.2005.04.054
Mustafa, Z: A new structure for generalized metric spaces with applications to fixed point theory. PhD Thesis, University of Newcastle, Australia (2005)
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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DOI: https://doi.org/10.1186/16871812201334