 Research
 Open Access
 Published:
Gβψcontractive type mappings in Gmetric spaces
Fixed Point Theory and Applications volume 2013, Article number: 123 (2013)
Abstract
In this paper, we introduce Gβψcontractive mappings which are generalizations of αψcontractive mappings in the context of Gmetric spaces. Additionally, we prove existence and uniqueness of fixed points of such contractive mappings. Our results generalize, extend and improve the existing results in the literature. We state some examples to illustrate our results.
1 Introduction and preliminaries
In the last few decades, fixed point theory has been one of the most interesting research fields in nonlinear functional analysis. In addition to many branches of applied and pure mathematics, fixed point theory results have wide application areas in many disciplines such as economics, computer science, engineering etc. The most remarkable results in this direction were given by Banach [1] in 1922. He proved that each contraction in a complete metric space has a unique fixed point. Due to application potential of the theory, many authors have directed their attention to this field and have generalized the Banach fixed point theorem in various ways (see, e.g., [1–50]). Very recently, Samet et al. [38] introduced the notion of αψcontractive mappings and proved the related fixed point theorems. The authors [38] showed that Banach fixed point theorems and some other theorems in the literature became direct consequences of their results. On the other hand, in 2004, Mustafa and Sims [24] defined the notion of a Gmetric space and characterized the Banach fixed point theorem in the context of a Gmetric space. Following these results, many authors have discussed fixed point theorems in the framework of Gmetric spaces; see, e.g., [9, 10, 14–16, 18–20, 23–29, 40–43, 50]. In this paper, we combine these two notions by introducing Gβψcontractive mappings, which are a characterization of αψcontractive mappings in the context of Gmetric spaces. Our main results generalize, extend and improve the existing results on the topic in the literature.
Throughout this paper, ℕ denotes the set of nonnegative integers, and {\mathbb{R}}^{+} denotes the set of nonnegative reals.
Let Ψ be a family of functions \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) satisfying the following conditions:

(i)
ψ is nondecreasing;

(ii)
there exist {k}_{0}\in \mathbb{N} and a\in (0,1) and a convergent series of nonnegative terms {\sum}_{k=1}^{\mathrm{\infty}}{v}_{k} such that
{\psi}^{k+1}(t)\le a{\psi}^{k}(t)+{v}_{k}
for k\ge {k}_{0} and any t\in {\mathbb{R}}^{+}.
These functions are known in the literature as (c)comparison functions.
Lemma 1 (See [5])
If \psi \in \mathrm{\Psi}, then the following hold:

(i)
{({\psi}^{n}(t))}_{n\in \mathbb{N}} converges to 0 as n\to \mathrm{\infty} for all t\in {\mathbb{R}}^{+};

(ii)
\psi (t)<t for any t\in (0,\mathrm{\infty});

(iii)
ψ is continuous at 0;

(iv)
the series {\sum}_{k=1}^{\mathrm{\infty}}{\psi}^{k}(t) converges for any t\in {\mathbb{R}}^{+}.
Remark 2 In some sources, (c)comparison functions are called BianchiniGrandolfi gauge functions (see, e.g., [34–36]).
Very recently, Samet et al. [38] introduced the following concepts.
Definition 3 Let (X,d) be a metric space and let T:X\to X be a given mapping. We say that T is an αψcontractive mapping if there exist two functions \alpha :X\times X\to [0,\mathrm{\infty}) and \psi \in \mathrm{\Psi} such that
for all x,y\in X.
Clearly, any contractive mapping, that is, a mapping satisfying Banach contraction, is an αψcontractive mapping with \alpha (x,y)=1 for all x,y\in X and \psi (t)=kt, for all t\ge 0 and some k\in [0,1).
Definition 4 Let T:X\to X and \alpha :X\times X\to [0,\mathrm{\infty}). We say that T is αadmissible if for all x,y\in X, we have
Various examples of such mappings are presented in [38]. The main results in [38] are the following fixed point theorems.
Theorem 5 Let (X,d) be a complete metric space and T:X\to X be an αψcontractive mapping. Suppose that

(i)
T is αadmissible;

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

(iii)
T is continuous.
Then there exists u\in X such that Tu=u.
Theorem 6 Let (X,d) be a complete metric space and T:X\to X be an αψcontractive mapping. Suppose that

(i)
T is αadmissible;

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

(iii)
if \{{x}_{n}\} is a sequence in X such that \alpha ({x}_{n},{x}_{n+1})\ge 1 for all n and {x}_{n}\to x\in X as n\to \mathrm{\infty}, then \alpha ({x}_{n},x)\ge 1 for all n.
Then there exists u\in X such that Tu=u.
Theorem 7 Adding to the hypotheses of Theorem 5 (resp. Theorem 6) the condition: For all x,y\in X, there exists z\in X such that \alpha (x,z)\ge 1 and \alpha (y,z)\ge 1, we obtain the uniqueness of a fixed point of T.
Mustafa and Sims [24] introduced the concept of Gmetric spaces as follows.
Definition 8 ([24])
Let X be a nonempty set and G:X\times X\times X\to {\mathbb{R}}^{+} be a function satisfying the following properties:

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

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

(G3)
G(x,x,y)\le G(x,y,z) for all 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 all x,y,z,a\in X (rectangle inequality).
Then the function G is called a generalized metric, or, more specifically, a Gmetric on X, and the pair (X,G) is called a Gmetric space.
Every Gmetric on X defines a metric {d}_{G} on X by
Example 9 Let (X,d) be a metric space. The function G:X\times X\times X\to {\mathbb{R}}^{+}, defined as either
or
for all x,y,z\in X, is a Gmetric on X.
Definition 10 ([24])
Let (X,G) be a Gmetric space, and let \{{x}_{n}\} be a sequence of points of X. We say that \{{x}_{n}\} is Gconvergent to x\in X if
that is, for any \epsilon >0, there exists N\in \mathbb{N} such that G(x,{x}_{n},{x}_{m})<\epsilon for all n,m\ge N. We call x the limit of the sequence and write {x}_{n}\to x or \underset{n\to \mathrm{\infty}}{lim}{x}_{n}=x.
Proposition 11 ([24])
Let (X,G) be a Gmetric space. The following are equivalent:

(1)
G({x}_{n},{x}_{n},x)\to 0 as n\to \mathrm{\infty};

(2)
G({x}_{n},x,x)\to 0 as n\to \mathrm{\infty};

(3)
G({x}_{n},{x}_{m},x)\to 0 as n,m\to \mathrm{\infty}.
Definition 12 ([24])
Let (X,G) be a Gmetric space. A sequence \{{x}_{n}\} is called a GCauchy sequence if for any \epsilon >0, there is N\in \mathbb{N} such that G({x}_{n},{x}_{m},{x}_{l})<\epsilon for all n,m,l\ge N, that is, G({x}_{n},{x}_{m},{x}_{l})\to 0 as n,m,l\to \mathrm{\infty}.
Proposition 13 ([24])
Let (X,G) be a Gmetric space. Then the following are equivalent:

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

(2)
for any \epsilon >0, there exists N\in \mathbb{N} such that G({x}_{n},{x}_{m},{x}_{m})<\epsilon for all n,m\ge N.
Definition 14 ([24])
A Gmetric space (X,G) is called Gcomplete if every GCauchy sequence is Gconvergent in (X,G).
Lemma 15 ([24])
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).
Definition 16 (See [24])
Let (X,G) be a Gmetric space. A mapping T:X\to X is said to be Gcontinuous if \{T({x}_{n})\} is Gconvergent to T(x), where \{{x}_{n}\} is a Gconvergent sequence converging to x.
In [23], Mustafa characterized the wellknown Banach contraction principle mapping in the context of Gmetric spaces in the following ways.
Theorem 17 (See [23])
Let (X,G) be a complete Gmetric space and let T:X\to X be a mapping satisfying the following condition for all x,y,z\in X:
where k\in [0,1). Then T has a unique fixed point.
Theorem 18 (See [23])
Let (X,G) be a complete Gmetric space and let T:X\to X be a mapping satisfying the following condition for all x,y\in X:
where k\in [0,1). Then T has a unique fixed point.
Remark 19 The condition (1) implies the condition (2). The converse is true only if k\in [0,\frac{1}{2}). For details, see [23].
From [23, 24], each Gmetric G on X generates a topology {\tau}_{G} on X whose base is a family of open Gballs \{{B}_{G}(x,\epsilon ):x\in X,\epsilon >0\}, where {B}_{G}(x,\epsilon )=\{y\in X:G(x,y,y)<\epsilon \} for all x\in X and \epsilon >0. A nonempty set A in the Gmetric space (X,G) is Gclosed if \overline{A}=A. Moreover,
Proposition 20 Let (X,G) be a Gmetric space and let A be a nonempty subset of X. The set A is Gclosed if for any Gconvergent sequence \{{x}_{n}\} in A with limit x, one has x\in A.
2 Main results
We introduce the concept of generalized αψcontractive mappings as follows.
Definition 21 Let (X,G) be a Gmetric space and let T:X\to X be a given mapping. We say that T is a Gβψcontractive mapping of type I if there exist two functions \beta :X\times X\times X\to [0,\mathrm{\infty}) and \psi \in \mathrm{\Psi} such that for all x,y,z\in X, we have
Definition 22 Let (X,G) be a Gmetric space and let T:X\to X be a given mapping. We say that T is a Gβψcontractive mapping of type II if there exist two functions \beta :X\times X\times X\to [0,\mathrm{\infty}) and \psi \in \mathrm{\Psi} such that for all x,y\in X, we have
Definition 23 Let (X,G) be a Gmetric space and let T:X\to X be a given mapping. We say that T is a Gβψcontractive mapping of type A if there exist two functions \beta :X\times X\times X\to [0,\mathrm{\infty}) and \psi \in \mathrm{\Psi} such that for all x,y\in X, we have
Remark 24 Clearly, any contractive mapping, that is, a mapping satisfying (1), is a Gβψcontractive mapping of type I with \beta (x,y,z)=1 for all x,y,z\in X and \psi (t)=kt, k\in (0,1). Analogously, a mapping satisfying (2), is a Gβψcontractive mapping of type II with \beta (x,y,y)=1 for all x,y\in X and \psi (t)=kt, k\in (0,1).
Definition 25 Let T:X\to X and \beta :X\times X\times X\to [0,\mathrm{\infty}). We say that T is βadmissible if for all x,y,z\in X, we have
Example 26 Let X=[0,\mathrm{\infty}) and T:X\to X. Define \beta (x,y,z):X\times X\times X\to [0,\mathrm{\infty}) by
Then T is βadmissible.
Our first result is the following.
Theorem 27 Let (X,G) be a complete Gmetric space. Suppose that T:X\to X is a Gβψcontractive mapping of type A and satisfies the following conditions:

(i)
T is βadmissible;

(ii)
there exists {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1;

(iii)
T is Gcontinuous.
Then there exists u\in X such that Tu=u.
Proof Let {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1 (such a point exists from the condition (ii)). Define the sequence \{{x}_{n}\} in X by {x}_{n+1}=T{x}_{n} for all n\ge 0. If {x}_{{n}_{0}}={x}_{{n}_{0}+1} for some {n}_{0}, then u={x}_{{n}_{0}} is a fixed point of T. So, we can assume that {x}_{n}\ne {x}_{n+1} for all n. Since T is βadmissible, we have
Inductively, we have
From (5) and (6), it follows that for all n\ge 1, we have
Since ψ is nondecreasing, by induction, we have
Using (G5) and (7), we have
Since \psi \in \mathrm{\Psi} and G({x}_{0},{x}_{1},{x}_{1})>0, by Lemma 1, we get
Thus, we have
By Proposition 13, this implies that \{{x}_{n}\} is a GCauchy sequence in the Gmetric space (X,G). Since (X,G) is complete, there exists u\in X such that \{{x}_{n}\} is Gconvergent to u. Since T is Gcontinuous, it follows that \{T{x}_{n}\} is Gconvergent to Tu. By the uniqueness of the limit, we get u=Tu, that is, u is a fixed point of T. □
The next theorem does not require continuity.
Theorem 28 Let (X,G) be a complete Gmetric space. Suppose that T:X\to X is a Gβψcontractive mapping of type A and satisfies the following conditions:

(i)
T is βadmissible;

(ii)
there exists {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1;

(iii)
if \{{x}_{n}\} is a sequence in X such that \beta ({x}_{n},{x}_{n+1},{x}_{n+1})\ge 1 for all n and \{{x}_{n}\} is a Gconvergent to x\in X, then \beta ({x}_{n},x,{x}_{n+1})\ge 1 for all n.
Then there exists u\in X such that Tu=u.
Proof Following the proof of Theorem 29, we know that the sequence \{{x}_{n}\} defined by {x}_{n+1}=T{x}_{n} for all n\ge 0 is a GCauchy sequence in the complete Gmetric space (X,G) that is Gconvergent to u\in X. From (6) and (iii), we have
Using the basic properties of Gmetric together with (5) and (8), we have
Letting n\to \mathrm{\infty}, using Proposition 11 and since ψ is continuous at t=0, it follows that
By Lemma 15, we obtain u=Tu. □
The following theorem can be derived easily from Theorems 27 and 28.
Theorem 29 Let (X,G) be a complete Gmetric space. Suppose that T:X\to X is a Gβψcontractive mapping of type A and satisfies the following conditions:

(i)
T is βadmissible;

(ii)
there exists {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1;

(iii)
T is Gcontinuous.
Then there exists u\in X such that Tu=u.
Theorem 30 Let (X,G) be a complete Gmetric space. Suppose that T:X\to X is a Gβψcontractive mapping of type II and satisfies the following conditions:

(i)
T is βadmissible;

(ii)
there exists {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1;

(iii)
if \{{x}_{n}\} is a sequence in X such that \beta ({x}_{n},{x}_{n+1},{x}_{n+1})\ge 1 for all n and \{{x}_{n}\} is a Gconvergent to x\in X, then \beta ({x}_{n},x,{x}_{n+1})\ge 1 for all n.
Then there exists u\in X such that Tu=u.
Remark 31 We notice that some fixed point theorems in the context of Gmetric space can be derived from usual fixed point results via certain substitutions (see, e.g., [21, 39]). On the other hand, our main result cannot be obtained via a substitution technique because the expressions in our statements do not allow one to achieve a metric by writing a simple substitution.
With the following example, we show that the hypotheses in Theorems 2730 do not guarantee uniqueness.
Example 32 Let X=[0,\mathrm{\infty}) be the Gmetric space, where
for all x,y\in X. Consider the selfmapping T:X\to X given by
Notice that Theorem 18 in [23], a characterization of the Banach fixed point theorem, cannot be applied in this case because G(T1,T2,T2)=4>2=G(1,2,2).
Define \beta :X\times X\times X\to [0,\mathrm{\infty}) as
Let \psi (t)=\frac{t}{2} for t\ge 0. Then we conclude that T is a Gβψcontractive mapping. In fact, for all x,y\in X, we have
On the other hand, there exists {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1. Indeed, for {x}_{0}=1, we have \beta (1,T1,T1)=\beta (1,\frac{1}{4},\frac{1}{4})=1.
Notice also that T is continuous. To show that T satisfies all the hypotheses of Theorem 29, it is sufficient to observe that T is βadmissible. For this purpose, let x,y\in X such that \beta (x,y,y)\ge 1, which is equivalent to saying that x,y\in [0,1]. Due to the definitions of β and T, we have
Hence, \beta (Tx,Ty,Ty)\ge 1. As a result, all the conditions of Theorem 29 are satisfied. Note that Theorem 29 guarantees the existence of a fixed point but not the uniqueness. In this example, 0 and \frac{7}{4} are two fixed points of T.
In the following example, T is not continuous.
Example 33 Let X, G and β be defined as in Example 32. Let T:X\to X be a map given by
Let \psi (t)=\frac{t}{3} for t\ge 0. Then we conclude that T is a Gβψcontractive mapping. In fact, for all x,y\in X, we have
Furthermore, there exists {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1. For {x}_{0}=1, we have \beta (1,T1,T1)=\beta (1,\frac{1}{3},\frac{1}{3})=1.
Let \{{x}_{n}\} be a sequence such that \beta ({x}_{n},{x}_{n+1},{x}_{n+1})\ge 1 for all n\in \mathbb{N} and {x}_{n}\to x as n\to \mathrm{\infty}. By the definition of β, we have \beta ({x}_{n},{x}_{n+1},{x}_{n+1})\ge 1 for all n\in \mathbb{N}. Then we see that {x}_{n}\in [0,1]. Thus, \beta ({x}_{n},x,x)\ge 1.
To show that T satisfies all of the hypotheses of Theorem 30, it is sufficient to observe that T is βadmissible. For this purpose, let x,y\in X such that \beta (x,y,y)\ge 1. It is equivalent to saying that x,y\in [0,1]. Due to the definitions of β and T, we have
Hence, \beta (Tx,Ty,Ty)\ge 1.
As a result, all the conditions of Theorem 30 are satisfied. Note that Theorem 30 guarantees the existence of a fixed point but not uniqueness. In this example, 0 and \frac{7}{4} are two fixed points of T.
Theorem 34 Adding the following condition to the hypotheses of Theorem 27 (resp. Theorem 29Theorem 30) we obtain the uniqueness of a fixed point of T.

(iv)
For all x,y\in X, there exists z\in X such that \beta (x,z,z)\ge 1 and \beta (y,z,z)\ge 1.
Proof Let u,{u}^{\ast}\in X be two fixed points of T. By (iv), there exists z\in X such that
Since T is βadmissible, we get by induction that
From (9) and (5), we have
Thus, we get by induction that
By (G4), we get
Letting n\to \mathrm{\infty}, and since \psi \in \mathrm{\Psi}, we have
This implies that \{{T}^{n}z\} is Gconvergent to u. Similarly, we get \{{T}^{n}z\} is Gconvergent to {u}^{\ast}. By the uniqueness of the limit, we get u={u}^{\ast}, that is, the fixed point of T is unique. □
3 Consequences
3.1 Cyclic contraction
Now, we prove our results for cyclic contractive mappings in a Gmetric space.
Theorem 35 Let A, B be a nonempty Gclosed subset of a complete Gmetric (X,G) space, let Y=A\cup B, and let T:Y\to Y be a given selfmapping satisfying
If there exists a function \psi \in \mathrm{\Psi} such that
then T has a unique fixed point u\in A\cap B, that is, Tu=u.
Proof Notice that (Y,G) is a complete Gmetric space because A, B are closed subsets of a complete Gmetric space (X,G). We define \beta :X\times X\times X\to [0,\mathrm{\infty}) in the following way:
Due to the definition of β and assumption (11), we have
Hence, T is a Gβψcontractive mapping.
Let (x,y)\in Y\times Y such that \beta (x,y,y)\ge 1. If (x,y)\in A\times B, then by assumption (10), (Tx,Ty)\in B\times A, which yields that \beta (Tx,Ty,Ty)\ge 1. If (x,y)\in B\times A, we get again \beta (Tx,Ty,Ty)\ge 1 by analogy. Thus, in any case we have \beta (Tx,Ty,Ty)\ge 1, that is, T is βadmissible. Notice also that for any z\in A, we have (z,Tz)\in A\times B, which yields that \beta (z,Tz,Tz)\ge 1.
Take a sequence \{{x}_{n}\} in X such that \beta ({x}_{n},{x}_{n+1},{x}_{n+1})\ge 1 for all n and {x}_{n}\to z\in X as n\to \mathrm{\infty}. Regarding the definition of β, we derive that
By the assumption, A, B and (A\times B)\cup (B\times A) are closed sets. Hence we get that (z,z)\in (A\times B)\cup (B\times A), which implies that z\in A\cap B. We conclude, by the definition of β, that \beta ({x}_{n},z,z)\ge 1 for all n.
Now all the hypotheses of Theorem 30 are satisfied, and we conclude that T has a fixed point. Next, we show the uniqueness of a fixed point z of T. Suppose that w=Tw, where w\in A\cap B. Since (z,w)(A\times B)\cup (B\times A), we have \beta (y,z,z)\ge 1 and \beta (z,y,y)\ge 1. Thus the condition (iv) of Theorem 34 is satisfied. □
3.2 Coupled fixed point theorems
For the rest of the paper, we suppose that all Gmetric spaces (X,G) are symmetric, that is, G(x,y,y)=G(x,x,y) for all x,y\in X.
In 1987, Guo and Lakshmikantham [8] introduced the notion of a coupled fixed point. The concept of a coupled fixed point was reconsidered by GnanaBhaskar and Lakshmikantham [7] in 2006. In this paper, they proved the existence and uniqueness of a coupled fixed point of an operator F:X\times X\to X on a partially ordered metric space under a condition called the mixed monotone property.
Definition 36 ([7])
Let (X,\u2aaf) be a partially ordered set and F:X\times X\to X. The mapping F is said to have the mixed monotone property if F(x,y) is monotone nondecreasing in x and monotone nonincreasing in y, that is, for any x,y\in X,
and
Definition 37 ([7])
An element (x,y)\in X\times X is called a coupled fixed point of the mapping F:X\times X\to X if
Lemma 38 (See [38])
Let F:X\times X\to X be a given mapping. Define the mapping {T}_{F}:X\times X\to X\times X by {T}_{F}(x,y)=(F(x,y),F(y,x)) for all (x,y)\in X\times X. Then (x,y) is a fixed point of {T}_{F} if and only if (x,y) is a coupled fixed point of F.
Definition 39 Let (X,G) be a Gmetric space. A mapping F:X\times X\to X is said to be continuous if for any two Gconvergent sequences \{{x}_{n}\} and \{{y}_{n}\} converging to x and y, respectively, \{F({x}_{n},{y}_{n})\} is Gconvergent to F(x,y).
Theorem 40 Let (X,G) be a complete Gmetric space and let F:X\times X\to X be a given mapping. Suppose there exist \psi \in \mathrm{\Psi} and a function \beta :{X}^{2}\times {X}^{2}\times {X}^{2}\to [0,\mathrm{\infty}) such that
for all (x,y),(u,v)\in X\times X. Suppose also that

(a)
for all (x,y),(u,v)\in X\times X, we have
\begin{array}{r}\beta ((x,y),(u,v),(u,v))\ge 1\\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\beta ((F(x,y),F(y,x)),(F(u,v),F(v,u)),(F(u,v),F(v,u)))\ge 1;\end{array} 
(b)
there exists ({x}_{0},{y}_{0}) such that
\begin{array}{c}\beta (({x}_{0},{y}_{0}),(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0})),(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0})))\ge 1,\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\hfill \\ \beta ((F({y}_{0},{x}_{0}),F({x}_{0},{y}_{0})),(F({y}_{0},{x}_{0}),F({x}_{0},{y}_{0})),({y}_{0},{x}_{0}))\ge 1;\hfill \end{array} 
(c)
F is continuous.
Then F has a coupled fixed point, that is, there exists ({x}^{\ast},{y}^{\ast})\in X\times X such that F({x}^{\ast},{y}^{\ast})={x}^{\ast} and F({y}^{\ast},{x}^{\ast})={y}^{\ast}.
Proof Let (Y,\delta ) be a complete Gmetric space with Y=X\times X and
for all (x,y),(u,v),(s,t)\in Y. By using (14) and (G4), we get
and
Combining (15) and (16), we have
for all \zeta =(x,y),\eta =(u,v)\in Y, where T:Y\to Y is defined by
and \gamma :Y\times Y\times Y\to [0,\mathrm{\infty}) is given by
It follows that T is a Gcontinuous and Gγψcontractive mapping of type II.
Suppose that \gamma (\zeta ,\eta ,\eta )\ge 1 for \zeta =(x,y),\eta =(u,v)\in Y. Then, by the condition (a), we have \gamma (T\zeta ,T\eta ,T\eta )\ge 1. Therefore, T is γadmissible.
From the condition (b), there exists ({x}_{0},{y}_{0}) such that
Since all the hypotheses of Theorem 29 are satisfied, it follows that T has a fixed point, and by Lemma 38, F has a coupled fixed point. □
Theorem 41 Let (X,G) be a complete Gmetric space and let F:X\times X\to X be a given mapping. Suppose there exist \psi \in \mathrm{\Psi} and a function \beta :{X}^{2}\times {X}^{2}\times {X}^{2}\to [0,\mathrm{\infty}) such that
for all (x,y),(u,v)\in X\times X. Suppose also that

(a)
for all (x,y),(u,v)\in X\times X, we have
\begin{array}{r}\beta ((x,y),(u,v),(u,v))\ge 1\\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\beta ((F(x,y),F(y,x)),(F(u,v),F(v,u)),(F(u,v),F(v,u)))\ge 1;\end{array} 
(b)
there exists ({x}_{0},{y}_{0}) such that
\begin{array}{c}\beta (({x}_{0},{y}_{0}),(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0})),(F({x}_{0},{y}_{0}),F({y}_{0},{x}_{0}))\ge 1,\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\hfill \\ \beta ((F({y}_{0},{x}_{0}),F({x}_{0},{y}_{0})),(F({y}_{0},{x}_{0}),F({x}_{0},{y}_{0})),({y}_{0},{x}_{0}))\ge 1;\hfill \end{array} 
(c)
if \{{x}_{n}\} and \{{y}_{n}\} are sequences in X such that
\beta (({x}_{n},{y}_{n}),({x}_{n+1},{y}_{n+1}),({x}_{n+1},{y}_{n+1}))\ge 1
and
\{{x}_{n}\} and \{{y}_{n}\} are Gconvergent to x and y, respectively, then
and
for all n.
Then F has a coupled fixed point, that is, there exists ({x}^{\ast},{y}^{\ast})\in X\times X such that F({x}^{\ast},{y}^{\ast})={x}^{\ast} and F({y}^{\ast},{x}^{\ast})={y}^{\ast}.
Proof Let \{({x}_{n},{y}_{n})\} be a sequence in Y such that
and ({x}_{n},{y}_{n}) is Gconvergent to (x,y). From the condition (c), we get
This implies that all the hypotheses of Theorem 30 are satisfied. It follows that T has a fixed point, and by Lemma 38, the mapping F has a coupled fixed point. □
Theorem 42 Adding the following condition to the hypotheses of Theorem 40 (resp. Theorem 41), we obtain the uniqueness of a coupled fixed point of F.

(d)
For all (x,y),(u,v)\in X\times X, there exists ({z}_{1},{z}_{2})\in X\times X such that
\beta ((x,y),({z}_{1},{z}_{2}),({z}_{1},{z}_{2}))\ge 1,\phantom{\rule{2em}{0ex}}\beta (({z}_{2},{z}_{1}),({z}_{2},{z}_{1}),(y,x))\ge 1
and
Proof With the condition (d), T and γ satisfy the condition (iv) of Theorem 34. From Theorem 34 and Lemma 38, the result follows. □
3.3 Choudhury and Maity’s coupled fixed point results in a Gmetric space
Definition 43 Let (X,\u2aaf) be a partially ordered set, and let (X,G) be a Gmetric space. A partially ordered Gmetric space, (X,G,\u2aaf), is called ordered complete if for each convergent sequence {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}\subset X, the following conditions hold:
(OC_{1}) if \{{x}_{n}\} is a nonincreasing sequence in X such that {x}_{n}\to {x}^{\ast}, then {x}^{\ast}\u2aaf{x}_{n} \mathrm{\forall}n\in \mathbb{N};
(OC_{2}) if \{{y}_{n}\} is a nondecreasing sequence in X such that {y}_{n}\to {y}^{\ast}, then {y}^{\ast}\u2ab0{y}_{n} \mathrm{\forall}n\in \mathbb{N}.
Choudhury and Maity [10] proved the following coupled fixed point theorems on ordered Gmetric spaces.
Theorem 44 Let (X,\u2aaf) be a partially ordered set and let G be a Gmetric on X such that (X,G) is a complete Gmetric space. Let F:X\times X\to X be a Gcontinuous mapping having the mixed monotone property on X. Suppose that there exists a k\in [0,1) such that
for all x,y,u,v,w,z\in X with x\u2ab0u\u2ab0w and y\u2aafv\u2aafz, where either u\ne w or v\ne z. If there exist {x}_{0},{y}_{0}\in X such that {x}_{0}\u2aafF({x}_{0},{y}_{0}) and F({y}_{0},{x}_{0})\u2aaf{y}_{0}, then F has a coupled fixed point, that is, there exists (x,y)\in X\times X such that x=F(x,y) and y=F(y,x).
Proof Let Y={X}^{2}. Suppose that \beta ,\gamma :Y\times Y\times Y\to [0,\mathrm{\infty}) such that
where
From (17), for all (x,y),(u,v)\in X\times X, we have
and
It follows that T is a Gγψcontractive mapping of type II with \psi (t)=kt, t\ge 0. Let (x,y),(u,v)\in X\times X such that
By the definition of γ, we get x\u2ab0u and y\u2aafv. This implies that
since F has the mixed monotone property. Thus,
By the assumption, there exist {x}_{0},{y}_{0}\in X such that {x}_{0}\u2aafF({x}_{0},{y}_{0}) and F({y}_{0},{x}_{0})\u2aaf{y}_{0}. By the definition of γ, it implies that
where s={x}_{0} and t={y}_{0}. From Theorem 40, F has a coupled fixed point. □
Theorem 45 If, instead of Gcontinuity of F in the theorem above, we assume that X is ordered complete, then F has a coupled fixed point.
Proof It is sufficient to prove that the condition (c) of Theorem 41 is satisfied under the setting of (18). For this purpose, we take two sequences \{{s}_{n}\} and \{{t}_{n}\} in X such that {s}_{n}\to s\in X and {t}_{n}\to t\in X as n\to \mathrm{\infty}. Assume that \beta (({s}_{n},{t}_{n}),({s}_{n+1},{t}_{n+1}),({s}_{n+1},{t}_{n+1}))\ge 1 and \beta (({t}_{n+1},{s}_{n+1}),({t}_{n+1},{s}_{n+1}),({t}_{n},{s}_{n}))\ge 1. Due to the definition of β, the sequences \{{s}_{n}\} and \{{t}_{n}\} are nonincreasing and nondecreasing, respectively. Regarding (i) and (ii), we derive that
which yields that
Then, the assumption (c) of Theorem 41 holds. Hence, F has a coupled fixed point. □
Remark 46 Notice that analogs of all of the theorems proved in Section 2 can be derived by replacing type I and type II with type A.
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales. Fundam. Math. 1922, 3: 133–181.
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
Agarwal RP, Alghamdi MA, Shahzad N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 40
Agarwal R, Karapınar E: Remarks on some coupled fixed point theorems in G metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 2
Berinde V: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare; 2002.
Ćirić L, Lakshmikantham V: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020
GnanaBhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
Guo D, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal., Theory Methods Appl. 1987, 11: 623–632. 10.1016/0362546X(87)900770
Aydi H, Damjanović B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered G metric spaces. Math. Comput. Model. 2011, 54: 2443–2450. 10.1016/j.mcm.2011.05.059
Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036
Karapınar E: Fixed point theory for cyclic weak ϕ contraction. Appl. Math. Lett. 2011, 24(6):822–825. 10.1016/j.aml.2010.12.016
Karapınar E, Sadaranagni K: Fixed point theory for cyclic ( ϕ  ψ )contractions. Fixed Point Theory Appl. 2011., 2011: Article ID 69
Karapınar E: Coupled fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl. 2010, 59: 3656–3668.
Aydi H, Karapınar E, Shatanawi W: Tripled fixed point results in generalized metric spaces. J. Appl. Math. 2012., 2012: Article ID 314279
Aydi H, Karapınar E, Mustafa Z: On common fixed points in G metric spaces using (E.A) property. Comput. Math. Appl. 2012, 64(6):1944–1956.
Aydi H, Karapınar E, Shatanawi W: Tripled common fixed point results for generalized contractions in ordered generalized metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 101
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
Karapınar E, Kaymakcalan B, Tas K: On coupled fixed point theorems on partially ordered G metric spaces. J. Inequal. Appl. 2012., 2012: Article ID 200
Ding HS, Karapınar E: A note on some coupled fixed point theorems on G metric space. J. Inequal. Appl. 2012., 2012: Article ID 170
Gul U, Karapınar E: On almost contraction in partially ordered metric spaces via implicit relation. J. Inequal. Appl. 2012., 2012: Article ID 217
Jleli M, Samet B: Remarks on G metric spaces and fixed point theorems. Fixed Point Theory Appl. 2012., 2012: Article ID 210
Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 2003, 4(1):79–89.
Mustafa, Z: A new structure for generalized metric spaces with applications to fixed point theory. Ph.D. thesis, The University of Newcastle, Australia (2005)
Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7(2):289–297.
Mustafa Z, Aydi H, Karapınar E: On common fixed points in imagemetric spaces using (E.A) property. Comput. Math. Appl. 2012. doi:10.1016/j.camwa.2012.03.051
Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mapping on complete G metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 189870
Mustafa Z, Khandaqji M, Shatanawi W: Fixed point results on complete G metric spaces. Studia Sci. Math. Hung. 2011, 48: 304–319.
Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175
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
Nashine HK, Sintunavarat W, Kumam P: Cyclic generalized contractions and fixed point results with applications to an integral equation. Fixed Point Theory Appl. 2012., 2012: Article ID 217
Pacurar M, Rus IA: Fixed point theory for cyclic φ contractions. Nonlinear Anal. 2010, 72: 1181–1187. 10.1016/j.na.2009.08.002
Petric MA: Some results concerning cyclical contractive mappings. Gen. Math. 2010, 18(4):213–226.
Petruşel A, Rus IA: Fixed point theorems in ordered L spaces. Proc. Am. Math. Soc. 2006, 134: 411–418.
Proinov PD: A generalization of the Banach contraction principle with high order of convergence of successive approximations. Nonlinear Anal., Theory Methods Appl. 2007, 67: 2361–2369. 10.1016/j.na.2006.09.008
Bianchini RM, Grandolfi M: Transformazioni di tipo contracttivo generalizzato in uno spazio metrico. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 1968, 45: 212–216.
Proinov PD: New general convergence theory for iterative processes and its applications to Newton Kantorovich type theorems. J. Complex. 2010, 26: 3–42. 10.1016/j.jco.2009.05.001
Rus IA: Cyclic representations and fixed points. Ann. ’Tiberiu Popoviciu’ Sem. Funct. Equ. Approx. Convexity 2005, 3: 171–178.
Samet B, Vetro C, Vetro P: Fixed point theorem for α  ψ contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014
Samet B, Vetro C, Vetro F: Remarks on G metric spaces. Int. J. Anal. 2013., 2013: Article ID 917158
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
Shatanawi W: Coupled fixed point theorems in generalized metric spaces. Hacet. J. Math. Stat. 2011, 40(3):441–447.
Shatanawi W, Abbas M, Nazir T: Common coupled coincidence and coupled fixed point results in two generalized metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 80
Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 81
Sintunavarat W, Cho YJ, Kumam P: Coupled fixedpoint theorems for contraction mapping induced by cone ballmetric in partially ordered spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 128
Sintunavarat W, Kumam P: Common fixed point theorem for cyclic generalized multivalued contraction mappings. Appl. Math. Lett. 2012, 25(11):1849–1855. 10.1016/j.aml.2012.02.045
Sintunavarat W, Kumam P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 93
Sintunavarat W, Petruşel A, Kumam P:Common coupled fixed point theorems for {w}^{\ast}compatible mappings without mixed monotone property. Rend. Circ. Mat. Palermo 2012, 61: 361–383. 10.1007/s1221501200960
Sintunavarat W, Radenović S, Golubović Z, Kumam P: Coupled fixed point theorems for F invariant set. Appl. Math. Inf. Sci. 2013, 7(1):247–255. 10.12785/amis/070131
Tahat N, Aydi H, Karapınar E, Shatanawi W: Common fixed points for singlevalued and multivalued maps satisfying a generalized contraction in G metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 48
Acknowledgements
The authors express their gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions. The authors also thanks to Professor İlker Savas Yüce for his help for improving the presentation of the paper. The first author supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Alghamdi, M.A., Karapınar, E. Gβψcontractive type mappings in Gmetric spaces. Fixed Point Theory Appl 2013, 123 (2013). https://doi.org/10.1186/168718122013123
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122013123
Keywords
 Fixed Point Theorem
 Contractive Mapping
 Fixed Point Theory
 Unique Fixed Point
 Couple Fixed Point