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MeirKeeler type contractions in partially ordered Gmetric spaces
Fixed Point Theory and Applications volume 2013, Article number: 35 (2013)
Abstract
In this paper, we establish several fixed point theorems for MeirKeeler type contractions in partially ordered Gmetric spaces.
MSC:46N40, 47H10, 54H25, 46T99.
1 Introduction and preliminaries
There are three main motivations for this paper. The first is the introduction of the concept of a Gmetric space and fixed point theorems on Gmetric spaces. The second is the works on fixed point theorems of MeirKeeler type contractions. The third is some recent works on fixed point theorems in a partially ordered set.
In this paper, we will combine these ideas and present some new results. In fact, due to the powerfulness of the classical Banach contraction principle in nonlinear analysis, various generalizations of the classical Banach contraction principle have been of great interest for many authors (see, e.g., [1–26]). Next, let us recall some definitions and known results.
In 2004, Mustafa and Sims [15] introduced the concept of Gmetric spaces as follows.
Definition 1 (See [15])
Let X be a nonempty set, 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 2 Let (X,d) be a metric space. The function G:X\times X\times X\to [0,+\mathrm{\infty}), defined by
or
for all x,y,z\in X, is a Gmetric on X.
Definition 3 (See [15])
Let (X,G) be a Gmetric space, and let \{{x}_{n}\} be a sequence of points of X, therefore, we say that ({x}_{n}) is Gconvergent to x\in X if {lim}_{n,m\to +\mathrm{\infty}}G(x,{x}_{n},{x}_{m})=0, 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 4 (See [15])
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 5 (See [15])
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 m,n,l\ge N, that is, G({x}_{n},{x}_{m},{x}_{l})\to 0 as n,m,l\to +\mathrm{\infty}.
Proposition 6 (See [15])
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 m,n\ge N.
Definition 7 (See [15])
A Gmetric space (X,G) is called Gcomplete if every GCauchy sequence is Gconvergent in (X,G).
Definition 8 (See [15])
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.
Definition 9 Let (X,\u2aaf) be a partially ordered set, (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}.
In [14], Mustafa characterized the wellknown Banach contraction principle mapping in the context of Gmetric spaces in the following ways.
Theorem 10 (See [14])
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 11 (See [14])
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 12 The condition (1.2) implies the condition (1.3). The converse is true only if k\in [0,\frac{1}{2}). For details, see [14].
Ran and Reurings [22] proved the analog of the Banach contraction mapping principle for continuous selfmappings under certain conditions in the context of a partially ordered set. In this paper [22], the authors solved a matric equation as an application. Following this initial paper, Nieto and López [20] published the paper in which the authors extended the results of Ran and Reurings [22] for a mapping T not necessarily continuous by assuming an additional hypothesis on (X,\u2aaf,d).
An interesting and general contraction condition for selfmaps in metric spaces was considered by Meir and Keeler [13] in 1969.
Definition 13 Let (X,d) be a metric space and T be a selfmap on X. Then T is called a MeirKeeler type contraction whenever for each \epsilon >0 there exists \delta >0 such that for any x,y\in X,
Recently, Harjani, Lopez and Sadarangani [7] extended the classical result in [13] to partially ordered metric spaces. In fact, they proved several interesting results for fixed points of MeirKeeler contractions in a complete metric space endowed with a partial order. For more related results, we refer the reader to [9, 10, 25] and references therein. Following this line of thought, we introduce a generalized MeirKeeler type contraction on Gmetric spaces and extend the results of [7, 13] in the context of partially ordered Gmetric spaces.
We say that the tripled (x,y,z)\in {X}^{3} is distinct if at least one of the following holds:
The tripled (x,y,z)\in {X}^{3} is called strictly distinct if all inequalities (i)(iii) hold.
Definition 14 Let (X,G,\u2aaf) be a partially ordered Gmetric space. Suppose that T:X\to X is a selfmapping satisfying the following condition:
For each \epsilon >0, there exists \delta >0 such that for any x,y,z\in X with x\u2aafy\u2aafz,
Then T is called GMeirKeeler contractive.
Remark 15 Notice that if T:X\to X is GMeirKeeler contractive on a Gmetric space (X,G), then T is contractive, that is,
for all distinct tripled (x,y,z)\in {X}^{3} with x\u2aafy\u2aafz.
Definition 16 Let (X,\u2aaf) be a partially ordered set and T:X\to X be a mapping. We say that T is nondecreasing if for x,y\in X,
Definition 17 Let (X,G,\u2aaf) be a Gmetric space. Suppose that T:X\to X is a selfmapping satisfying the following condition:
Given \epsilon >0, there exists \delta >0 such that for any x,y\in X with x\u2aafy,
Then T is called GMeirKeeler contractive of second type.
Remark 18 It is easy to see that a GMeirKeeler contraction must be GMeirKeeler contractive of second type. In addition, if T:X\to X is GMeirKeeler contractive of second type on a partially ordered Gmetric space (X,G,\u2aaf), then
for all (x,y)\in {X}^{2} with x\prec y. Moreover, we have
for all (x,y)\in {X}^{2} with x\u2aafy.
2 Main results
In this paper, we discuss the existence of fixed points for a MeirKeeler type contraction in partially ordered Gmetric spaces.
Theorem 19 Let (X,\u2aaf) be a partially ordered set endowed with a Gmetric and T:X\to X be a given mapping. Suppose that the following conditions hold:

(i)
(X,G) is Gcomplete;

(ii)
T is nondecreasing (with respect to ⪯);

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

(iv)
T is Gcontinuous;

(v)
T:X\to X is GMeirKeeler contractive of second type.
Then T has a fixed point. Moreover, if for all (x,y)\in X\times X there exists w\in X such that x\u2aafw and y\u2aafw, we obtain the uniqueness of the fixed point.
Proof The following proof follows the same lines as previous proofs of related results in [7, 13], but we reproduce it for the sake of completeness. More precisely, the first part of the proof for Theorem 19, the proof up to equation (2.13), is analogous to the corresponding proof of Harjani et al. in [7]. But, for the general readership, we give all the details here.
Take {x}_{0}\in X such that the condition (iii) holds, that is, {x}_{0}\u2aafT{x}_{0}. We construct an iterative sequence \{{x}_{n}\} in X as follows:
Taking into account that T is a nondecreasing mapping together with (2.1), we have {x}_{0}\u2aafT{x}_{0}={x}_{1} implies {x}_{1}=T{x}_{0}\u2aafT{x}_{1}={x}_{2}. By induction, we get
Suppose that there exists {n}_{0} such that {x}_{{n}_{0}}={x}_{{n}_{0}+1}. Since {x}_{{n}_{0}}={x}_{{n}_{0}+1}=T{x}_{{n}_{0}}, then {x}_{{n}_{0}} is the fixed point of T, which completes the existence part of the proof. Suppose that {x}_{n}\ne {x}_{n+1} for all n\in \mathbb{N}. Thus, by (2.2) we have
By (G2), we have
for all n=0,1,2,\dots . By Remark 18, we observe that for all n=0,1,2,\dots ,
Define {t}_{n}=G({x}_{n},{x}_{n+1},{x}_{n+1}). Due to (2.5), the sequence \{{t}_{n}\} is a (strictly) decreasing sequence in {\mathbb{R}}^{+} and thus it is convergent, say t\in {\mathbb{R}}^{+}. We claim that t=0. Suppose, to the contrary, that t>0. Thus, we have
Assume \epsilon =t>0. Then by hypothesis, there exists a convenient \delta (\epsilon )>0 such that (1.8) holds. On the other hand, due to the definition of ε, there exists {n}_{0}\in \mathbb{N} such that
Taking the condition (1.8) into account, the expression (2.7) yields that
which contradicts (2.6). Hence t=0, that is, {lim}_{n\to \mathrm{\infty}}{t}_{n}=0.
We will show that {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} is a GCauchy sequence. \mathrm{\forall}\epsilon >0, by the hypothesis, there exists a suitable \delta (\epsilon )>0 such that (1.8) holds. Without loss of generality, we assume \delta <\epsilon. Since t=0, there exists {N}_{0}\in \mathbb{N} such that
We assert that for any fixed k\ge {N}_{0},
holds. To prove the assertion, we use the method of induction. Regarding (2.9), the assertion (2.10) is satisfied for r=1. Suppose the assertion (2.10) is satisfied for r=1,2,\dots ,m for some m\in \mathbb{N}. For r=m+1, by the help of (G5) and (2.9), we consider
If G({x}_{k1},{x}_{k+m},{x}_{k+m})\ge \epsilon, then by (1.8) we get
Hence (2.10) is satisfied.
If G({x}_{k1},{x}_{k+m},{x}_{k+m})=0, then by (G2), we derive that {x}_{k1}={x}_{k+m} and hence {x}_{k}=T{x}_{k1}=T{x}_{k+m}={x}_{k+m+1}. By (G1), we have
and thus (2.10) is satisfied.
If 0<G({x}_{k1},{x}_{k+m},{x}_{k+m})<\epsilon, then by Remark 18,
Consequently, (2.10) is satisfied for r=m+1. Hence, G({x}_{k},{x}_{k+r},{x}_{k+r})\le \epsilon for all k\ge {N}_{0} and r\ge 1, which means
Then, for all n\ge m\ge {N}_{0}, by (2.13), we have
Thus, for all m,n\ge {N}_{0}, there holds
By Proposition 6, \{{x}_{n}\} is a GCauchy sequence. Since (X,G) is Gcomplete, there exists u\in X such that
We will show now that u\in X is a fixed point of T, that is, u=Tu. Since T is Gcontinuous, the sequence \{T{x}_{n}\}=\{{x}_{n+1}\} converges to Tu, that is,
On the other hand, the rectangle inequality (G5) yields that
Letting n\to \mathrm{\infty} in (2.16), we conclude that G(u,Tu,Tu)=0. Hence, u=Tu, that is, u∈ is a fixed point of T.
To prove the uniqueness, we assume that v\in X is another fixed point of T. By the assumptions, we know that there exists w\in X such that u\u2aafw and v\u2aafw. By Remark 18, we get
Since T is nondecreasing, Tu\u2aafTw. Again by Remark 18, we get
Continuing in this way, we conclude
Let {s}_{n}=G(u,{T}^{n}w,{T}^{n}w). Hence, we conclude that \{{s}_{n}\} is a nonincreasing sequence bounded below by zero. Thus, there exists L\ge 0 such that
We claim that L=0. Suppose, on the contrary, that L>0. Choose \epsilon =L and \delta >0 be such that (1.8) holds. Then, there exists {n}_{0} such that L\le G(u,{T}^{{n}_{0}}w,{T}^{{n}_{0}}w)<L+\delta, which implies
This contradicts with the definition of L. Hence,
Similarly, one can also obtain
In view of (2.17), (2.18) and
we deduce G(u,v,v)=0, i.e., u=v. Hence, the fixed point of T is unique. □
Corollary 20 Let (X,\u2aaf) be a partially ordered set endowed with a Gmetric and T:X\to X be a given mapping. Suppose that the following conditions hold:

(i)
(X,G) is Gcomplete;

(ii)
T is nondecreasing (with respect to ⪯);

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

(iv)
T is Gcontinuous;

(v)
T:X\to X is GMeirKeeler contractive.
Then T has a fixed point. Moreover, if for all (x,y)\in X\times X, there exists w\in X such that x\u2aafw and y\u2aafw, we obtain the uniqueness of the fixed point.
Substituting the condition (iv) in Theorem 19 by the condition that X is ordered complete, we can get the following result.
Theorem 21 Let (X,\u2aaf) be a partially ordered set endowed with a Gmetric and T:X\to X be a given mapping. Suppose that the following conditions hold:

(i)
(X,G) is Gcomplete;

(ii)
T is nondecreasing (with respect to ⪯);

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

(iv)
X is ordered complete;

(v)
T:X\to X is GMeirKeeler contractive of second type.
Then T has a fixed point. Moreover, if for all (x,y)\in X\times X there exists w\in X such that x\u2aafw and y\u2aafw, we obtain the uniqueness of the fixed point.
Proof Let {x}_{n} and u be as in the proof of Theorem 19. We only need to show u=Tu. Since X is ordered complete, in view of (2.2) and (2.14), we conclude {x}_{n}\u2aafu for all n. Then, by Remark 18, (G5) and (2.14), we get
Letting n\to \mathrm{\infty}, we conclude G(Tu,u,u)=0, i.e., Tu=u. □
Corollary 22 Let (X,\u2aaf) be a partially ordered set endowed with a Gmetric and T:X\to X be a given mapping. Suppose that the following conditions hold:

(i)
(X,G) is Gcomplete;

(ii)
T is nondecreasing (with respect to ⪯);

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

(iv)
X is ordered complete;

(v)
T:X\to X is GMeirKeeler contractive.
Then T has a fixed point. Moreover, if for all (x,y)\in X\times X there exists w\in X such that x\u2aafw and y\u2aafw, we obtain the uniqueness of the fixed point.
Theorem 23 Let (X,\u2aaf) be a partially ordered set endowed with a Gmetric and T:X\to X be a given mapping. Suppose that there exists a function \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) satisfying the following conditions

(F1)
\phi (0)=0 and \phi (t)>0 for all t>0;

(F2)
φ is nondecreasing and right continuous;

(F3)
for every \epsilon >0, there exists δ such that
\epsilon \le \phi (G(x,y,y))<\epsilon +\delta \phantom{\rule{1em}{0ex}}\mathit{\text{implies}}\phantom{\rule{1em}{0ex}}\phi (G(Tx,Ty,Ty))<\phi (\epsilon )(2.19)
for all (x,y)\in X\times X with x\u2aafy. Then T is GMeirKeeler contractive of second type.
Proof We take \epsilon >0. Due to (F1), we have \phi (\epsilon )>0. Thus there exists \theta >0 such that
From the right continuity of φ, there exists \delta >0 such that \phi (\epsilon +\delta )<\phi (\epsilon )+\theta. Fix (x,y)\in X\times X with x\u2aafy such that \epsilon \le G(x,y,y)<\epsilon +\delta. So, we have
Hence, \phi (G(Tx,Ty,Ty))<\phi (\epsilon ). Thus, we have G(Tx,Ty,Ty)<\epsilon, which completes the proof. □
Since a function t\to {\int}_{0}^{t}f(s)\phantom{\rule{0.2em}{0ex}}ds is absolutely continuous, we derive the following corollary from Theorem 23 and Theorem 19.
Corollary 24 Let (X,\u2aaf) be a partially ordered set endowed with a Gmetric, T:X\to X be a given mapping, and f be a locally integrable function from [0,\mathrm{\infty}) into itself satisfying {\int}_{0}^{t}f(s)\phantom{\rule{0.2em}{0ex}}ds>0 for all t>0. Assume that the conditions (i)(iv) of Theorem 19 hold, and for each \epsilon >0, there exists \delta >0 such that
for all x,y\in X with x\u2aafy. Then T has a fixed point. Moreover, if for all (x,y)\in X\times X, there exists w\in X such that x\u2aafw and y\u2aafw, we obtain the uniqueness of the fixed point.
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Acknowledgements
The authors thank the referees for their valuable comments that helped to improve the text. HuiSheng Ding acknowledges support from the NSF of China, and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University.
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Ding, HS., Karapınar, E. MeirKeeler type contractions in partially ordered Gmetric spaces. Fixed Point Theory Appl 2013, 35 (2013). https://doi.org/10.1186/16871812201335
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DOI: https://doi.org/10.1186/16871812201335