Abstract
In this paper, we introduce the notion of generalized Gβψ contractive mappings which is inspired by the concept of αψ contractive mappings. We showed the existence and uniqueness of a fixed point for such mappings in the setting of complete Gmetric spaces. The main results of this paper extend, generalize and improve some wellknown results on the topic in the literature. We state some examples to illustrate our results. We consider also some applications to show the validity of our results.
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1 Introduction and preliminaries
In nonlinear functional analysis, the importance of fixed point theory has been increasing rapidly as an interesting research field. One of the most important reasons for this development is the potential of application of fixed point theory not only in various branches of applied and pure mathematics, but also in many other disciplines such as chemistry, biology, physics, economics, computer science, engineering etc. We also emphasize the crucial role of celebrated results of Banach [1], known as a Banach contraction mapping principle or a Banach fixed point theorem, in the growth of this theory. In 1922, Banach proved that every contraction in a complete metric space has a unique fixed point. After this remarkable paper, a number of authors have extended/generalized/improved the Banach contraction mapping principle in various ways in different abstract spaces (see, e.g., [2–22]). One of the interesting and recent results in this direction was given by Samet et al. [23]. They defined the notion of αψ contractive mappings and proved that including the Banach fixed point theorems, some wellknown fixed point results turn into corollaries of their results. Another interesting result was given in 2004 by Mustafa and Sims [24] by introducing the notion of a Gmetric space as a generalization of the concept of a metric space. The authors characterized the Banach fixed point theorem in the context of a Gmetric space. After this result, many authors have paid attention to this space and proved the existence and uniqueness of a fixed point in the context of a Gmetric space (see, e.g., [11, 17–20, 24–48]). In this paper, we combine these two notions by introducing a Gβψ contractive mapping which is a characterization αψ contractive mappings in the context of Gmetric spaces. Our main results generalize, extend and improve the existence results on the topic in the literature.
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}}^{+}, where {\mathbb{R}}^{+}=[0,\mathrm{\infty}).
These functions are known in the literature as BianchiniGrandolfi gauge functions in some sources (see, e.g., [21, 22, 49]) and as (c)comparison functions in some other sources (see, e.g., [50]).
Lemma 1 (See [50])
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 {\mathbb{R}}^{+};

(iii)
ψ is continuous at 0;

(iv)
the series {\sum}_{k=1}^{\mathrm{\infty}}{\psi}^{k}(t) converges for any t\in {\mathbb{R}}^{+}.
Very recently, Karapınar and Samet [32] introduced the following concepts.
Definition 2 Let (X,d) be a metric space and T:X\to X be a given mapping. We say that T is a generalized αψ contractive 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, where
Clearly, since ψ is nondecreasing, every αψ contractive mapping, presented in [23], is a generalized αψ contractive mapping.
Definition 3 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 [23]. The main results in [32] are the following fixed point theorems.
Theorem 4 Let (X,d) be a complete metric space and T:X\to X be a generalized αψ contractive 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 5 Let (X,d) be a complete metric space and T:X\to X be a generalized αψ contractive 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 6 Adding to the hypotheses of Theorem 4 (resp. Theorem 5) the condition: For all x,y\in Fix(T), there exists z\in X such that \alpha (x,z)\ge 1 and \alpha (y,z)\ge 1, we obtain the uniqueness of the fixed point of T.
Mustafa and Sims [24] introduced the concept of Gmetric spaces as follows.
Definition 7 [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 8 Let (X,d) be a metric space. The function G:X\times X\times X\to {\mathbb{R}}^{+}, defined as
or
for all x,y,z\in X, is a Gmetric on X.
Definition 9 [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 {lim}_{n\to \mathrm{\infty}}{x}_{n}=x.
Proposition 10 [24]
Let (X,G) be a Gmetric space. The following are equivalent:

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

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

(3)
G({x}_{n},x,x)\to 0 as n\to \mathrm{\infty};

(4)
G({x}_{n},{x}_{m},x)\to 0 as n,m\to \mathrm{\infty}.
Definition 11 [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 12 [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 13 [24]
A Gmetric space (X,G) is called Gcomplete if every GCauchy sequence is Gconvergent in (X,G).
Lemma 14 [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 15 (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 any Gconvergent sequence converging to x.
In [36], Mustafa characterized the wellknown Banach contraction principle mapping in the context of Gmetric spaces in the following way.
Theorem 16 (See [36])
Let (X,G) be a complete Gmetric space and 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 17 (See [36])
Let (X,G) be a complete Gmetric space and 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 18 The condition (1) implies the condition (2). The converse is true only if k\in [0,\frac{1}{2}). For details, see [36].
From [24, 36], 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. Moreover,
Proposition 19 Let (X,G) be a Gmetric space and A be a nonempty subset of X. Then 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 αψ contractive mappings as follows.
Definition 20 Let (X,G) be a Gmetric space and T:X\to X be a given mapping. We say that T is a generalized Gβψ contractive 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
where
Definition 21 Let (X,G) be a Gmetric space and T:X\to X be a given mapping. We say that T is a generalized Gβψ contractive 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
where
Remark 22 Clearly, any contractive mapping, that is, a mapping satisfying (1), is a generalized Gβψ contractive 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 generalized Gβψ contractive mapping of type II with \beta (x,y,y)=1 for all x,y\in X and \psi (t)=kt, where k\in (0,1).
Definition 23 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 24 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 Tx=ln(1+x) and
Then T is βadmissible.
Our first result is the following.
Theorem 25 Let (X,G) be a complete Gmetric space. Suppose that T:X\to X is a generalized Gβψ contractive mapping of type I and satisfies the following conditions:
(i)_{ a } T is βadmissible;
(ii)_{ a } there exists {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1;
(iii)_{ b } T is Gcontinuous.
Then there exists u\in X such that Tu=u.
Proof Let {x}_{0}\in X be such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1 (such a point exists from the condition (ii)_{ a }). 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 (3) and (5), it follows that for all n\ge 1, we have
On the other hand, we have
Thus, we have
We consider the following two cases:
Case 1: If max\{G({x}_{n1},{x}_{n},{x}_{n}),G({x}_{n},{x}_{n+1},{x}_{n+1})\}=G({x}_{n},{x}_{n+1},{x}_{n+1}) for some n, then
which is a contradiction.
Case 2: If max\{G({x}_{n1},{x}_{n},{x}_{n}),G({x}_{n},{x}_{n+1},{x}_{n+1})\}=G({x}_{n1},{x}_{n},{x}_{n}), then
for all n\ge 1. Since ψ is nondecreasing, by induction, we get
Using (G5) and (6), we have
Since \psi \in \mathrm{\Psi} and G({x}_{0},{x}_{1},{x}_{1})>0, by Lemma 1, we get that
Thus, we have
By Proposition 12, 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. □
Definition 26 (See [51])
Let (X,G) be a Gmetric space and T:X\to X be a given mapping. We say that T is a Gβψ contractive 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
by following the lines of the proof of Theorem 25.
Corollary 27 Let (X,G) be a complete Gmetric space. Suppose that T:X\to X is a Gβψ contractive mapping of type I and satisfies the following conditions:
(i)_{ a } T is βadmissible;
(ii)_{ a } there exists {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1;
(iii)_{ b } T is Gcontinuous.
Then there exists u\in X such that Tu=u.
Example 28 Let X=[0,\mathrm{\infty}) be endowed with the Gmetric
Define T:X\to X by Tx=3x for all x\in X. We define \beta :X\times X\times X\to [0,\mathrm{\infty}) in the following way:
One can easily show that
Then T is a Gβψ contractive mapping of type I with \psi (t)=\frac{1}{9}t for all t\in [0,\mathrm{\infty}). Take x,y,z\in X such that \beta (x,y,z)\ge 1. By the definition of T, this implies that x=y=z=0. Then we have \beta (Tx,Ty,Tz)=\beta (0,0,0)=1, and so T is βadmissible. All the conditions of Corollary 27 are satisfied. Here, 0 is the fixed point of T. Notice also that the Banach contraction mapping principle is not applicable. Indeed, d(x,y)=xy for all x,y\in X. Then we have x\ne y d(Tx,Ty)=3xy>kxy for all k\in [0,1).
It is clear that Theorem 16 is not applicable.
The following result can be easily concluded from Theorem 25.
Corollary 29 Let (X,G) be a complete Gmetric space. Suppose that T:X\to X is a generalized Gβψ contractive mapping of type II and satisfies the following conditions:
(i)_{ a } T is βadmissible;
(ii)_{ a } there exists {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1;
(iii)_{ b } T is Gcontinuous.
Then there exists u\in X such that Tu=u.
The next theorem does not require the continuity of T.
Theorem 30 Let (X,G) be a complete Gmetric space. Suppose that T:X\to X is a generalized Gβψ contractive mapping of type I such that ψ is continuous and satisfies the following conditions:
(i)_{ b } T is βadmissible;
(ii)_{ b } there exists {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1;
(iii)_{ b } 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)\ge 1 for all n.
Then there exists u\in X such that Tu=u.
Proof Following the proof of Theorem 25, 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 (5) and (iii)_{ b }, we have
Using (8), we have
where
Letting n\to \mathrm{\infty} in the following inequality:
it follows that
which is a contradiction. Thus, we obtain G(u,Tu,Tu)=0, that is, by Lemma 14, u=Tu. □
The following corollary can be easily derived from Theorem 30.
Corollary 31 Let (X,G) be a complete Gmetric space. Suppose that T:X\to X is a generalized Gβψ contractive mapping of type II such that ψ is continuous and satisfies the following conditions:
(i)_{ b } T is βadmissible;
(ii)_{ b } there exists {x}_{0}\in X such that \beta ({x}_{0},T{x}_{0},T{x}_{0})\ge 1;
(iii)_{ b } 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)\ge 1 for all n.
Then there exists u\in X such that Tu=u.
With the following example, we will show that the hypotheses in Theorems 25 and 30 do not guarantee uniqueness.
Example 32 Let X=\{(1,0),(0,1)\}\subset {\mathbb{R}}^{2} be endowed with the following Gmetric:
for all (x,y),(u,v),(z,w)\in X. Obviously, (X,G) is a complete metric space. The mapping T(x,y)=(x,y) is trivially continuous and satisfies, for any \psi \in \mathrm{\Psi},
for all (x,y),(u,v),(z,w)\in X, where
Thus T is a generalized Gβψ contractive mapping. On the other hand, for all (x,y),(u,v),(z,w)\in X, we have
which yields that
Hence T is βadmissible. Moreover, for all (x,y)\in X, we have \beta ((x,y),T(x,y),T(x,y))\ge 1. So, the assumptions of Theorem 25 are satisfied. Note that the assumptions of Theorem 30 are also satisfied, indeed, if \{({x}_{n},{y}_{n})\} is a sequence in X that converges to some point (x,y)\in X with \beta (({x}_{n},{y}_{n}),({x}_{n+1},{y}_{n+1}),({x}_{n+1},{y}_{n+1}))\ge 1 for all n, then from the definition of β, we have ({x}_{n},{y}_{n})=(x,y) for all n, which implies that \beta (({x}_{n},{y}_{n}),(x,y),(x,y))=1 for all n. However, in this case, T has two fixed points in X.
Let X be a set and T be a selfmapping on X. The set of all fixed points of T will be denoted by Fix(T).
Theorem 33 Adding the following condition to the hypotheses of Theorem 25 (resp. Theorem 30, Corollary 29, Corollary 31), we obtain the uniqueness of the fixed point of T.

(iv)
For x\in Fix(T), \beta (x,z,z)\ge 1 for all z\in X.
Proof Let u,v\in Fix(T) be two fixed points of T. By (iv), we derive
Notice that \beta (Tu,Tv,Tv)=\beta (u,v,v) since u and v are fixed points of T. Consequently, we have
where
Thus, we get that
which is a contradiction. Therefore, u=v, i.e., the fixed point of T is unique. □
Corollary 34 Let (X,G) be a complete Gmetric space and let T:X\to X be a given mapping. Suppose that there exists a continuous function \psi \in \mathrm{\Psi} such that
for all x,y,z\in X. Then T has a unique fixed point.
Corollary 35 Let (X,G) be a complete Gmetric space and let T:X\to X be a given mapping. Suppose that there exists a function \psi \in \mathrm{\Psi} such that
for all x,y,z\in X. Then T has a unique fixed point.
Corollary 36 Let (X,G) be a complete Gmetric space and let T:X\to X be a given mapping. Suppose that there exists \lambda \in [0,1) such that
for all x,y,z\in X. Then T has a unique fixed point.
Corollary 37 Let (X,G) be a complete Gmetric space and let T:X\to X be a given mapping. Suppose that there exist nonnegative real numbers a, b, c, d and e with a+b+c+d+e<1 such that
for all x,y,z\in X. Then T has a unique fixed point.
Corollary 38 (See [40])
Let (X,G) be a complete Gmetric space and let T:X\to X be a given mapping. Suppose that there exists \lambda \in [0,1) such that
for all x,y,z\in X. Then T has a unique fixed point.
3 Consequences
3.1 Fixed point theorems on metric spaces endowed with a partial order
Definition 39 Let (X,\u2aaf) be a partially ordered set and T:X\to X be a given mapping. We say that T is nondecreasing with respect to ⪯ if
Definition 40 Let (X,\u2aaf) be a partially ordered set. A sequence \{{x}_{n}\}\subset X is said to be nondecreasing with respect to ⪯ if
Definition 41 Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X. We say that (X,\u2aaf,G) is Gregular if for every nondecreasing sequence \{{x}_{n}\}\subset X such that {x}_{n}\to x\in X as n\to \mathrm{\infty}, {x}_{n}\u2aafx for all n.
Theorem 42 Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X such that (X,G) is a complete Gmetric space. Let T:X\to X be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function \psi \in \mathrm{\Psi} such that
for all x,y\in X with x\u2aafy. Suppose also that the following conditions hold:

(i)
there exists {x}_{0}\in X such that {x}_{0}\u2aafT{x}_{0};

(ii)
T is Gcontinuous or (X,\u2aaf,G) is Gregular and ψ is continuous.
Then there exists u\in X such that Tu=u. Moreover, if for x\in Fix(T), x\u2aafz for all z\in X, one has the uniqueness of the fixed point.
Proof Define the mapping \beta :X\times X\times X\to [0,\mathrm{\infty}) by
From (9), for all x,y\in X, we have
It follows that T is a generalized Gβψ contractive mapping of type II. From the condition (i), we have
By the definition of β and since T is a nondecreasing mapping with respect to ⪯, we have
Thus T is βadmissible. Moreover, if T is Gcontinuous, by Theorem 25, T has a fixed point.
Now, suppose that (X,\u2aaf,G) is Gregular. Let \{{x}_{n}\} be a sequence in X such that \beta ({x}_{n},{x}_{n+1},{x}_{n+1})\ge 1 for all n and {x}_{n} is Gconvergent to x\in X. By Definition 41, {x}_{n}\u2aafx for all n, which gives us \beta ({x}_{n},x,x)\ge 1 for all k. Thus, all the hypotheses of Theorem 30 are satisfied and there exists u\in X such that Tu=u. To prove the uniqueness, since u\in Fix(T), we have, u\u2aafz for all z\in X. By the definition of β, we get that \beta (u,z,z)\ge 1 for all z\in X. Therefore, the hypothesis (iv) of Theorem 33 is satisfied and we deduce the uniqueness of the fixed point. □
Corollary 43 Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X such that (X,G) is a complete Gmetric space. Let T:X\to X be a nondecreasing mapping with respect to ⪯. Suppose that there exists a function \psi \in \mathrm{\Psi} such that
for all x,y\in X with x\u2aafy. Suppose also that the following conditions hold:

(i)
there exists {x}_{0}\in X such that {x}_{0}\u2aafT{x}_{0};

(ii)
T is Gcontinuous or (X,\u2aaf,G) is Gregular.
Then there exists u\in X such that Tu=u. Moreover, if for x\in Fix(T), x\u2aafz for all z\in X, one has the uniqueness of the fixed point.
Corollary 44 Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X such that (X,G) is a complete Gmetric space. Let T:X\to X be a nondecreasing mapping with respect to ⪯. Suppose that there exists \lambda \in [0,1) such that
for all x,y\in X with x\u2aafy. Suppose also that the following conditions hold:

(i)
there exists {x}_{0}\in X such that {x}_{0}\u2aafT{x}_{0};

(ii)
T is Gcontinuous or (X,\u2aaf,G) is Gregular.
Then there exists u\in X such that Tu=u. Moreover, if for x\in Fix(T), x\u2aafz for all z\in X, one has the uniqueness of the fixed point.
Corollary 45 Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X such that (X,G) is a complete Gmetric space. Let T:X\to X be a nondecreasing mapping with respect to ⪯. Suppose that there exist nonnegative real numbers a, b, c and d with a+b+c+d<1 such that
for all x,y\in X with x\u2aafy. Suppose also that the following conditions hold:

(i)
there exists {x}_{0}\in X such that {x}_{0}\u2aafT{x}_{0};

(ii)
T is Gcontinuous or (X,\u2aaf,G) is Gregular.
Then there exists u\in X such that Tu=u. Moreover, if for x\in Fix(T), x\u2aafz for all z\in X, one has the uniqueness of the fixed point.
Corollary 46 Let (X,\u2aaf) be a partially ordered set and G be a Gmetric on X such that (X,G) is a complete Gmetric space. Let T:X\to X be a nondecreasing mapping with respect to ⪯. Suppose that there exists a constant \lambda \in [0,1) such that
for all x,y\in X with x\u2aafy. Suppose also that the following conditions hold:

(i)
there exists {x}_{0}\in X such that {x}_{0}\u2aafT{x}_{0};

(ii)
T is Gcontinuous or (X,\u2aaf,G) is Gregular.
Then there exists u\in X such that Tu=u. Moreover, if for x\in Fix(T), x\u2aafz for all z\in X, one has the uniqueness of the fixed point.
3.2 Cyclic contraction
Now, we will prove our results for cyclic contractive mappings in a Gmetric space.
Let A, B be a nonempty Gclosed subset of a complete Gmetric space (X,G). Suppose also that Y=A\cup B and T:Y\to Y is a given selfmapping satisfying
If there exists a continuous 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 since A, B is a closed subset 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 the assumption (12), we have
Hence, T is a generalized Gβψ contractive mapping.
Let (x,y)\in Y\times Y be such that \beta (x,y,y)\ge 1. If (x,y)\in A\times B then by the assumption (11), (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 \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 u\in X as n\to \mathrm{\infty}. Regarding the definition of β, we derive that
By assumption, A, B and hence (A\times B)\cup (B\times A) is a Gclosed set. Hence, we get that (u,u)\in (A\times B)\cup (B\times A), which implies that u\in A\cap B. We conclude, by the definition of β, that \beta ({x}_{n},u,u)\ge 1 for all n.
Now, all hypotheses of Theorem 30 are satisfied and we conclude that T has a fixed point. Next, we show the uniqueness of a fixed point u of T. Since u\in Fix(T) and u\in A\cap B, we get \beta (u,a,a)\ge 1 for all a\in Y. Thus, the condition (iv) of Theorem 33 is satisfied. □
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Alghamdi, MA, Karapınar, E: Gβψ contractive type mappings and related fixed point theorems. Preprint
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The research of the first author was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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Alghamdi, M.A., Karapınar, E. Gβψ contractivetype mappings and related fixed point theorems. J Inequal Appl 2013, 70 (2013). https://doi.org/10.1186/1029242X201370
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DOI: https://doi.org/10.1186/1029242X201370