Abstract
In this paper, we establish several fixed point theorems for Meir-Keeler type contractions in partially ordered G-metric spaces.
MSC:46N40, 47H10, 54H25, 46T99.
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1 Introduction and preliminaries
There are three main motivations for this paper. The first is the introduction of the concept of a G-metric space and fixed point theorems on G-metric spaces. The second is the works on fixed point theorems of Meir-Keeler 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 G-metric spaces as follows.
Definition 1 (See [15])
Let X be a non-empty set, be a function satisfying the following properties:
-
(G1)
if ,
-
(G2)
for all with ,
-
(G3)
for all with ,
-
(G4)
(symmetry in all three variables),
-
(G5)
for all (rectangle inequality).
Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair is called a G-metric space.
Every G-metric on X defines a metric on X by
Example 2 Let be a metric space. The function , defined by
or
for all , is a G-metric on X.
Definition 3 (See [15])
Let be a G-metric space, and let be a sequence of points of X, therefore, we say that is G-convergent to if , that is, for any , there exists such that for all . We call x the limit of the sequence and write or .
Proposition 4 (See [15])
Let be a G-metric space. The following are equivalent:
-
(1)
is G-convergent to x,
-
(2)
as ,
-
(3)
as ,
-
(4)
as .
Definition 5 (See [15])
Let be a G-metric space. A sequence is called a G-Cauchy sequence if for any , there is such that for all , that is, as .
Proposition 6 (See [15])
Let be a G-metric space. Then the following are equivalent:
-
(1)
the sequence is G-Cauchy,
-
(2)
for any , there exists such that for all .
Definition 7 (See [15])
A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .
Definition 8 (See [15])
Let be a G-metric space. A mapping is said to be G-continuous if is G-convergent to where is any G-convergent sequence converging to x.
Definition 9 Let be a partially ordered set, be a G-metric space. A partially ordered G-metric space is called ordered complete if for each convergent sequence , the following conditions hold:
(OC1) if is a non-increasing sequence in X such that , then ,
(OC2) if is a non-decreasing sequence in X such that , then .
In [14], Mustafa characterized the well-known Banach contraction principle mapping in the context of G-metric spaces in the following ways.
Theorem 10 (See [14])
Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Theorem 11 (See [14])
Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Remark 12 The condition (1.2) implies the condition (1.3). The converse is true only if . For details, see [14].
Ran and Reurings [22] proved the analog of the Banach contraction mapping principle for continuous self-mappings 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 .
An interesting and general contraction condition for self-maps in metric spaces was considered by Meir and Keeler [13] in 1969.
Definition 13 Let be a metric space and T be a self-map on X. Then T is called a Meir-Keeler type contraction whenever for each there exists such that for any ,
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 Meir-Keeler 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 Meir-Keeler type contraction on G-metric spaces and extend the results of [7, 13] in the context of partially ordered G-metric spaces.
We say that the tripled is distinct if at least one of the following holds:
The tripled is called strictly distinct if all inequalities (i)-(iii) hold.
Definition 14 Let be a partially ordered G-metric space. Suppose that is a self-mapping satisfying the following condition:
For each , there exists such that for any with ,
Then T is called G-Meir-Keeler contractive.
Remark 15 Notice that if is G-Meir-Keeler contractive on a G-metric space , then T is contractive, that is,
for all distinct tripled with .
Definition 16 Let be a partially ordered set and be a mapping. We say that T is nondecreasing if for ,
Definition 17 Let be a G-metric space. Suppose that is a self-mapping satisfying the following condition:
Given , there exists such that for any with ,
Then T is called G-Meir-Keeler contractive of second type.
Remark 18 It is easy to see that a G-Meir-Keeler contraction must be G-Meir-Keeler contractive of second type. In addition, if is G-Meir-Keeler contractive of second type on a partially ordered G-metric space , then
for all with . Moreover, we have
for all with .
2 Main results
In this paper, we discuss the existence of fixed points for a Meir-Keeler type contraction in partially ordered G-metric spaces.
Theorem 19 Let be a partially ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is nondecreasing (with respect to ⪯);
-
(iii)
there exists such that ;
-
(iv)
T is G-continuous;
-
(v)
is G-Meir-Keeler contractive of second type.
Then T has a fixed point. Moreover, if for all there exists such that and , 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 such that the condition (iii) holds, that is, . We construct an iterative sequence in X as follows:
Taking into account that T is a non-decreasing mapping together with (2.1), we have implies . By induction, we get
Suppose that there exists such that . Since , then is the fixed point of T, which completes the existence part of the proof. Suppose that for all . Thus, by (2.2) we have
By (G2), we have
for all . By Remark 18, we observe that for all ,
Define . Due to (2.5), the sequence is a (strictly) decreasing sequence in and thus it is convergent, say . We claim that . Suppose, to the contrary, that . Thus, we have
Assume . Then by hypothesis, there exists a convenient such that (1.8) holds. On the other hand, due to the definition of ε, there exists such that
Taking the condition (1.8) into account, the expression (2.7) yields that
which contradicts (2.6). Hence , that is, .
We will show that is a G-Cauchy sequence. , by the hypothesis, there exists a suitable such that (1.8) holds. Without loss of generality, we assume . Since , there exists such that
We assert that for any fixed ,
holds. To prove the assertion, we use the method of induction. Regarding (2.9), the assertion (2.10) is satisfied for . Suppose the assertion (2.10) is satisfied for for some . For , by the help of (G5) and (2.9), we consider
If , then by (1.8) we get
Hence (2.10) is satisfied.
If , then by (G2), we derive that and hence . By (G1), we have
and thus (2.10) is satisfied.
If , then by Remark 18,
Consequently, (2.10) is satisfied for . Hence, for all and , which means
Then, for all , by (2.13), we have
Thus, for all , there holds
By Proposition 6, is a G-Cauchy sequence. Since is G-complete, there exists such that
We will show now that is a fixed point of T, that is, . Since T is G-continuous, the sequence converges to Tu, that is,
On the other hand, the rectangle inequality (G5) yields that
Letting in (2.16), we conclude that . Hence, , that is, u∈ is a fixed point of T.
To prove the uniqueness, we assume that is another fixed point of T. By the assumptions, we know that there exists such that and . By Remark 18, we get
Since T is nondecreasing, . Again by Remark 18, we get
Continuing in this way, we conclude
Let . Hence, we conclude that is a non-increasing sequence bounded below by zero. Thus, there exists such that
We claim that . Suppose, on the contrary, that . Choose and be such that (1.8) holds. Then, there exists such that , 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 , i.e., . Hence, the fixed point of T is unique. □
Corollary 20 Let be a partially ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is nondecreasing (with respect to ⪯);
-
(iii)
there exists such that ;
-
(iv)
T is G-continuous;
-
(v)
is G-Meir-Keeler contractive.
Then T has a fixed point. Moreover, if for all , there exists such that and , 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 be a partially ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is nondecreasing (with respect to ⪯);
-
(iii)
there exists such that ;
-
(iv)
X is ordered complete;
-
(v)
is G-Meir-Keeler contractive of second type.
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain the uniqueness of the fixed point.
Proof Let and u be as in the proof of Theorem 19. We only need to show . Since X is ordered complete, in view of (2.2) and (2.14), we conclude for all n. Then, by Remark 18, (G5) and (2.14), we get
Letting , we conclude , i.e., . □
Corollary 22 Let be a partially ordered set endowed with a G-metric and be a given mapping. Suppose that the following conditions hold:
-
(i)
is G-complete;
-
(ii)
T is nondecreasing (with respect to ⪯);
-
(iii)
there exists such that ;
-
(iv)
X is ordered complete;
-
(v)
is G-Meir-Keeler contractive.
Then T has a fixed point. Moreover, if for all there exists such that and , we obtain the uniqueness of the fixed point.
Theorem 23 Let be a partially ordered set endowed with a G-metric and be a given mapping. Suppose that there exists a function satisfying the following conditions
-
(F1)
and for all ;
-
(F2)
φ is nondecreasing and right continuous;
-
(F3)
for every , there exists δ such that
(2.19)
for all with . Then T is G-Meir-Keeler contractive of second type.
Proof We take . Due to (F1), we have . Thus there exists such that
From the right continuity of φ, there exists such that . Fix with such that . So, we have
Hence, . Thus, we have , which completes the proof. □
Since a function is absolutely continuous, we derive the following corollary from Theorem 23 and Theorem 19.
Corollary 24 Let be a partially ordered set endowed with a G-metric, be a given mapping, and f be a locally integrable function from into itself satisfying for all . Assume that the conditions (i)-(iv) of Theorem 19 hold, and for each , there exists such that
for all with . Then T has a fixed point. Moreover, if for all , there exists such that and , 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. Hui-Sheng 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. Meir-Keeler type contractions in partially ordered G-metric spaces. Fixed Point Theory Appl 2013, 35 (2013). https://doi.org/10.1186/1687-1812-2013-35
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DOI: https://doi.org/10.1186/1687-1812-2013-35